Learn Russian: Я ни у кого не просил коньяку!

This is the partitive genitive (sometimes called the second genitive). For certain uncountable masculine nouns (like drinks, foods, or materials), an -у or -ю ending is used instead of the standard -а or -я when referring to a portion or "some" of that thing. Because the speaker is talking about asking for some cognac, коньяку is used. Note that while коньяка is also grammatically correct in modern Russian, коньяку sounds much more natural for a portion.

Why is the preposition у right in the middle of the phrase ни у кого? Can I say у никого instead?

No, you cannot say у никого. In Russian, when a negative pronoun like никто (nobody) is used with a preposition (like у), the preposition splits the pronoun right down the middle. The prefix ни- breaks off, the preposition goes in the center, and the rest of the pronoun follows: ни + у + кого. It literally structures as "not from whom."

Why do we use the preposition у with the person being asked?

The verb просить (to ask) requires the preposition у plus the genitive case for the person you are making the request to. Think of it as asking to take something from someone's possession or domain (у кого).