He is for most a friendly benevolent person who likes to help others.
Hippocrates spent the majority of his early life in the secrecy and isolation of a Mage family. His family were "relic keepers" intrusted to guard over knowledge from the old Greek deities, especially Asclepius and Apollo in his families' case. But ever since his childhood days Hippocrates was always more interested in helping humans than the duties of a traditional Mage.
So after some journeying on his home island Kos he saw how insufficient the treatments of wounds and diseases were, as the only form of treatment that was at the time known was some medicinal herbs and primitive compressions of injuries. He then asked his father why they aren't helping them. With the Magecraft and knowledge they have it would be an simple matter to help them, Hippocrates innocently argued. But his father harshly slapped the boy across the cheek and rebuked him for uttering "such nonsense". Mages devoted themselves to uncovering the deeper truths of the world, he coldly stated. To immerse themselves in worldly affairs, even to save others, would only distract them from that grand purpose.
Hippocrates bit back a retort, pretending to obey his father's words. However that day a fury began to blossom deeply within his chest. A cold rage at those who'd hoard knowledge to themselves while leaving the ignorant masses to die. Steeling his heart Hippocrates came to a decision. Instead of researching his family's Magecraft in the actual sense he'd examine what parts of it would be relevant for him and discard the rest.
As he has basically left his position as heir of a sorcerer family he has towards traditional Mages a rather uncomfortable relationship. He views that knowledge should not be hoarded and be a privilege to few individuals but shared upon the whole mankind. But he is no fool or naïve individual. He knew that even if Enforcers or Executors didn't exist at his time in that form, throughout Greece and perhaps even in other countries relic keepers had many connections or money to hire mercenaries to preserve their ancient secrets. He did his best to conceal his familial background as well as coded the Humorism as "medical basic theory" to not make himself the target of the keeper families' wrath.
Thus he can potentially work quite well with non-traditional Spellcasters provided they aren't mercenaries as he has a strong dislike for that occupation. Though he denies to have made encounters to such people he has in life even treated soldiers and similar.
"I don't like this occupation as it's the counterpart to my own. They take and I at least try to save lives."
As he is a pacifist he prefers to be play a supportive role, reinforcing his Master while they engage in combat directly. Though if forced to participate actively in a fight it seems that he is no inexperienced fighter as he can quickly read the patterns of his opponent and find loopholes to take advantages. But in either case he fights not because he enjoys it, no, he fights to put an end to it. As such he attempts to hold the fights as short as possible, so if he knew that his Master is safe he would retreat immediately.
He openly doubts at the meaning of pursuing the Root. If brought in a conversation with his Master he will openly point out to think what consequences that can provoke especially considering that the World actively hunts down those who try to come near it which speaks that the Root itself holds either inconceivable power or danger of also unexplainable dimension.
Demeanor Towards Other Servants:
He is towards the other Servants in a Grail War neutral and friendly as he sees himself foremost as a physician. As such he commands a neutral stance at first but also due to his pacifistic nature he treats them with respect and praising as especially the warriors and kings under them have performed feats that were rewarded with ascension into a Heroic Spirit. Though he greatly loathes Assassin (and especially poisoners) type Servants as they gained their fame for acts of dishonor or malevolence. So he will if confronted with such a Servant actively pursue them and bring their crimes to an end and would even be willing to ally with other Servants that would seem necessary to confront them.
As he has vivid knowledge about the old Greek Heroes and Deities- even having some blood of them- he is naturally very thrilled if he happens to encounter them. Then he would draw pen and paper and act like an eager reporter (or fan?) and want to ask them various questions. Should he be summoned in an Great Grail War type scenario he would express though surprisingly great interest in the second Caster as he himself admits.
"To be honest I am not disinterested in the subject of Magecraft but only the secrecy of the Mages is tedious to me."
Demeanour towards his Master:
He is a very reliable Servant who supports his Master in an honest way, and strive to help them in their endeavors even if that means to reach the Root. Albeit he will be a critical advisor who will even hinder /interrupt them if he sees that their research might harm themselves. In general should his Master trust his advice as his mind birthed the idea of medicine so he is very able to deduce the development of a subject or theory and if the outcome might be dangerous for even the finder.
It goes without saying that he prefers a good natured Master over an evil one. He will not directly betray an evil Master but he know up to which certain point he can cooperate and on which it's better to sever his contract.
Aside from the Grail War he is strongly interested to learn about the newest medicinal results. He also expresses a great deal of curiosity in medicinal practices from beyond Europe. In particular he found Asia as a treasure-trove full of medical traditions like TCM, Ayurveda, Ji Gong and Shaolin which explains why he dons Asian clothes as he likes quite much the calmness and collected mental stance that favors deliberation over excessive ambition. He sees the fact that Asian humans had reached high ages regularly even before the modern age as evidence for their efficient healthcare traditions and with them harmonious lifestyles in comparison to the healthcare of medieval Europe which he sees as greatly insufficient and religious biased. But he is pleased with the present state of medicine.
Dialogues a la Chaldea:
Florence Nightingale:
He sees the Master cornered by the nurse who Points her gun at him. Shortly before she can pull the trigger he disarms her and hits her with his clipboard resolutely.
"Idiot, what do you think you're doing!?" As he says that does he corrodes her rifle via alchemy to make it unusable.
She responds
"Why are you interrupting me?!" and gives him a death glare at first but then she sees his terrifying radiation which normally is a warm and calming aura but if he notices incorrect medical circumstances he get a terrifying aura that would even scar the First Hassan, thus she is snapped out of her monomania.
He then explains in a calm but raging voice
"Because Health Point one: medics don't hurt their patients. You are causing the patient visible stress and discomfort which should be avoided and kept as minimal as possible. Point two:" then he takes a calming breath through which his aura is calmed down before continuing.
"Your medical knowledge and understanding is great but the treatment situation is equally important. If you cause stress to your patients they develop stress based psychological phobias that stand in the way of rehabilitation, understood?"
He values much her medical engagement and her deeds she made and make but he can't tolerate that she scar and intimidate her patients. In return she when confronted with him seems to regain for some moments her sanity and to be honest she has great respect as he founded the principles of medicine so she does not dare to argue with him as he in those respects has the emanation of a general.
Asclepius:
"Not only my ancestor but truly also the first physician in history. It's a great honor to meet you Asclepius!! How would it be if we over a cup of tea discuss our knowledge and the result of our discoveries? I'm especially interested in hearing your experience towards medical cases that contain supernatural elements!"
I think that their relationship should be obvious but if not they like to exchange their experiences, case studies etc. and both have a high opinion of each other.
Heracles:
"Heracles... The greatest man that the Herodotical world has produced! .....I am in awe to encounter you Master Heracles i..ha..*coughs*
Would you have something against if I ask you some questions?!?"
His maternal ancestor. Alone the thought that he has a drop of his blood flowing in him is exceptional for Hippocrates and he himself thinks that this drop of Heracles blood in him has protected him from the punishment of the Gods.
Epione:
"The wife of Asclepius and the mother of his children, which means that you are also my ancestress. I'm pleased to meet you *bows and gives her a hand-kiss*, your beauty is indeed proving your divine ancestry."
"Between ourselves dear lady I would like to tell you Lady Epione, I understand why you are angry at Asclepius. Although I can see through his actions in the sanctuary which demonstrate that he is a brilliant researcher I can see that he isn't only loving his work. I dare to say *he straightens his glasses* that he is history's first mad scientist so I would try to approach him on his job. I can easily make a position at the sanctuary free for you."
As she is like Asclepius himself an ancestress from him on the paternal side he has great respect and adoration for her. Thus if she should have need for help he is ready to offer her anytime assistance (on personal level beside his sense of duty as a physician).
As well he does after get her to know that while he has inherited Asclepius' brilliance as a researcher he thinks that her compassionate side may also have left its mark on him in combination of the boundary breaking potential of his maternal ancestor Heracles.
Mary Seacole:
"Don't mind that your fame was greatly taken by Florence. You made the lives of dozens of soldiers more bearable and, yes, she has enjoyed good medical teachings but you have inherited an important aspect which she forgotten. Namely to help the soul of the patients to ease their stress and give them good time and a homey atmosphere to enable them to regenerate through the soul to the body. So don't sweat it as I can say that alongside the other healers of history besides Nightingale also you Mary are within Órkos Tou Ippokráti."
He quite likes her as she has the right demeanor towards her patients. Namely to ensure that around the actual medical treatment the environment supports the rehabilitation process. As well he highly esteems her friendly demeanour.
Paracelsus:
"I feel that we both are quite similar as we both are coming from an established Mage family but wanted to share our knowledge with the rest of the humans instead of hoarding it. Though it seems that you made some errors that I was able to avoid but nevermind would you like to chat a while with me over a cup of tea, maybe could we invite Theophrastes to our round?"
As both were Mages that didn't want to hide their knowledge they get along very well as they exchange their knowledge. They are basically somewhat "brothers in spirit".
Theophrastes:
"You are the founder of Botany, the teaching of plantlife. I must admit that I examined plantlife only for their value as medicine and not the plant itself. But would you be okay with telling me some of your knowledge...?"
"What have you said!? You taught some of my script elements in your school!? Wo that makes me *he wholeheartly smiles and nearly laughs*
proud to hear so early."
Phidias(Lostbelt):
"This ….this was the dream that you have imagined Phidias?! I have misjudged you. I thought that you wanted to make the world more beautiful with your art that nurtures the spirit of humans, not to turn them into statues."
Deianira:
"The greatest wife of Heracles, it is a great honor to meet you! I don't know if the ancestors of my mother were your sons or those others from Heracles but nevertheless I am honored to meet an adventuress. Pease tell me, would you have something against that I ask you some questions?"
Semiramis:
"The first poisoner in history huh..! *makes an enraged face*
As I have even in my life treated some poisoned victims I don't trust people like you! Albeit I have sworn to help anyone I make for you an exception. So if you pardon me I have some patients that need my help..." *he then walks away*