For the “Tenkai” form, an amalgamation of the Houjutsushi and Mitsuhide's personalities due to the latter fusing with the former's soul. The Servant is akin to a “Wise Grandfather-figure”; willing to lend his knowledge, and isn’t above pointing out your faults with a steady hand. Most of his words of wisdom, however, are extremely cryptic thus often frustrating to understand. While listening to Tenkai’s advices may seem counterintuitive, when deciphered, nearly all of his advices actually help.
For a wise man, Tenkai has a strange way to convey ideas; if the abstract advices weren't enough, he does so in the form of some extremely bizarre anecdotes. Such oddities include: A man attempts to increase donation rates to shrines by granting the people's wishes, or a customer at an restaurant buying all the food so the employees will leave early. Somehow, at the end of each tale, he managed to place an important life lesson to learn from the story.
Despite his eccentrics, Tenkai can occasionally act somber; especially during conversations regarding human mortality, the cycle of war and peace, and the means one will stoop to maintain said peace. Occasionally he would converse about philosophy, and would remark on how so many people threw their lives away in the name of someone they don't really know, or how honor and loyalty can blind someone of the Greater Good...
If there's any faults about the monk, it would be his staunch belief that everything can be solved with words. No matter how low are the chances of success, Tenkai would still insist that he should calmly reason with the enemy. Unfortunately, this also applies during Holy Grail Wars, where he would always propose ideas of alliance and truce to everyone. Tenkai will even butt into fights just to encourage a ceasefire. However, should conflict continue, he will not hesitate to defend his Master and himself.
As for the “Mitsuhide” form, the Servant's original personality, is surprisingly a decent person. Although can be too smart for his own good, he is an compassionate man to the core, Mitsuhide will treat his Master like a childhood friend, and will protect them with his life. However, if the Master consistently order/perform morally ill actions, Mitsuhide will not hesitate to kill them. Even if it means losing the Grail War, Mitsuhide will not abide to a cruel master again.
Mitsuhide was far more educated than other officers in his life, although mostly was self-taught. Usually he’s caught reading a book on poetry, for literature being one of his favorite subjects. He’s very interested about the culture of the modern world, and while he’s rather saddened by the world’s mostly negative view of him, Mitsuhide doesn’t regret his decision for it shaped Japan as it is today.
While it's rare, Mitsuhide can be somewhat of a smartass. He usually can’t help but occasionally boast about his superior intellect to his Master, allies or even his enemies. Mitsuhide’s desire to not offend anyone was another one of his problems, as it made him look seemingly distant and false to other people. As a result, this made him difficult for other individuals to approach and befriend.
Mitsuhide's Innocent Monster Skill did not help matters either; not only were the episodes random, he tend to act like a completely different person, making him an unstable and potentially dangerous Servant for a Master to deal with.
Aside from his Master, both Mitsuhide and Tenkai places the lives of innocents as a number one priority, and will even flee a winning fight if it guarantees the safety of a bystander. And if Oda Nobunaga were to be present, both Mitsuhide and Tenkai will place the Demon King as their priority target and will stop at nothing to kill her. For if she were to win a Grail War, it would mean a living hell to the whole world.
Regarding a wish from the Grail, Tenkai isn’t really care for a wish, since he died without any regrets. As for Mitsuhide, however, it’s to wish that his daughter Gracia to live a safe and happy life instead of her tragic fate.
(NOTE: To compare the Servant's forms, Tenkai is an Older and Wiser(?) person who learned from his mistakes, while Mitsuhide is still a troubled individual still dealing with the consequences of his actions.)
(In truth, Tenkai giving riddle-like advices is simply his way of having fun and screwing with people. Said advices working out for them was simply a side benefit.)
(Just like pointing out flaws with a steady hand, Tenkai will not hesitate to call you out for your personal problems. However, he only does that when said flaw was becoming a liability.)