I am currently playing as Master Yi but based on research I guess that's wrong. I am looking for a new character to play. I do not like jungeling, I like battles with other champions and I like to push the fight and attack. I am looking for help on who to play, assassians appeal to me but I dunno about their viability. Please help me!

Jax is one of the better low cost ones. You should also try Xin and Wukong during free weeks. Renekton fits and is free this week but he's kind of fallen out of favor (I'm not sure why, I never played him).

Nasus, Garen, Jax, and Darius are one of my favorite melees, very fun and easy to play.

I like master yi and also like jax, riven, darius, and others. They can put out a ton of damage, are great brawlers and have some form of sticking mechanism. If you have issues with being kited, try nidalee, jayce, or yorick top. They can do great things as a melee top but also have good poke.

Katarina, Talon, and Kha'zix are more expensive options that are nevertheless great. They are melee assassins, so you'll be doing a lot of battles with other champions (okay, you're going to farm the other champions like you farm minions) with lots of burst damage.

I've never played Garen. But I know when the other team builds him "properly"(or as properly as Lv8s can) I fear him. I'd definitely say check him out.

Yi is fun to play he has high damage when you buy the right gear. Farm minions to get the money. I usually stick with the recommended items and I do pretty well it tells you what his recommended items are at the starting point. Also I played as amumu he is more of a tank but he can deal decent damage. I haven't played many other melee champs I am more of a ranged kinda guy. I use Ashe, Annie, Ryze, Sivir mainly but I do use Amumu and Yi when I get the urge to use them. Taric is okay he is more support but if you farm minions he can get pretty good toward higher levels plus his heal makes him nearly invincible unless you go up against a high powered jax or someone else.

Renekton can be a really powerful solo-top. With the Tiamat and lots of AD, his minion clearance speed is insanely fast, so that you can either create a push or hold one.