My personal favorite has always been a tie between the Scout and the Pyro.
Scout is very mobile and allows for aggressive playstyles.
Pyro can do a little bit of everything, has combos and he can really fuck you if you're using projectiles (except Syringes).
I default to picking Pyro when I need to pick my class on 1st connect but I like playing Heavy and Pyro in-general. I only play Dragon's Fury Pyro though as I think the predictions it takes to use it are versatile and fun.
My favorite class to play are soldier, heavy and sniper depending on the circumstances.
character wise its the heavy because of his iconic lines like:
"WHO TOUCHED MY GUN"
"SANDVICH MAKES ME STRONG"
"Ya-ta-ta-ta-ta-ta-ta-ta ya-ta-ta-ta-ta-ta-ta-ta do-de-da-va-da-da-dada! Kaboom-Kaboom!"