How can I evoke a sense of dislike/hate for a character through game mechanics?

You need not create explicit game mechanics for this. Indeed, doing so can lead to the player hating your game rather than your character.

A safer course is this one: simply design your game as normal. But every time you throw a new curve at the player, assign the blame for it to the badguy. Portal provides many examples:

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The Weighted Companion Cube certainly brought you good luck. However, it cannot accompany you through the testing sequence and, sadly, must be euthanized.

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General writing tips for how to make a character unlikable are more for Writers SE (If you decide to post there, make sure your question is not too broad). But because you posted this on game development stackexchange, I will try to focus on those methods which are unique to games and do not apply to any other medium.

A good way to make a player hate a character is to make that character interfere negatively with their gameplay in ways which are perceived as unfair and annoying.

• Have them steal their powerups and their resources (directly from them or before they can pick them up) and have them destroy what the player created.
• Have them create additional obstacles and challenges for the player which don't have any reward when overcome.
• When the character is a direct opponent, have them cheat. AIs cheating is something we do very often, but usually we try to hide it as best as we can. If you want to make the player hate the character, make it very obvious they are cheating.
• Have the player do things which are tedious, boring or humiliating while making it obvious that the character is the direct reason they have to do it.
• Have the character distract the player from something far more interesting.

Do not give the player a chance to defend against any of these.

Keep in mind that each of these techniques makes the experience of the game itself worse, so it carries a high risk that the players will not just hate the character but will also hate the whole game. So do not overdo it! Make sure each of these negative experiences is rather short and sandwich them between more positive game experiences (which you should attribute to other characters which you want to be more likable).