It depends on what lane you usually play.
can hard carry a team. He has decent mixed damage, a bit of CC, and good durability. However, that is only late game. Contrary to popular belief, Jax gets countered by a large percentage of top laners. Some of them he can overcome late game, but others such as Jayce have kits that can neutralize Jax's kit. (Jax can never get close to Jayce. Jayce can land free poke and knock Jax away.)
I played Jax to get to Gold before the Guinsoo buff, and it might work for you too if you are looking for someone to carry with.
mid is also incredibly powerful. Most mid laners don't expect to fight against Cho'Gath. His strong early game damage, sustain, and CC will not only allow him to snow ball himself, but also completely deprive his enemy of gold. Just don't be an idiot and build him full AP. Generally, get ROA, Frozen Heart, Spirit Visage, Liandrie's, Rylai's, and Merc Treads.
If you trust your team enough to not feed too much in the early game, you can also go JG.
It is probably the easiest to hard carry as a Jungler once you get ahead.
is the best champion to hard carry with. His learning curve is pretty mild. Just farm for 8 minutes straight and one shot their carries in a team fight.
is also very good. He can shut down the enemy jungler, and can secure objectives easily due to his boxes and clones.