Early-game, Zilean is an excellent harasser. Because his bombs are his only offensive spell, they have good scaling, which means he can do good damage with very low contact time. If you're jumping in & out and trading single spells, zilean tends to do well.
Time Warp is one of the best utility spells in the game. Used well in a team scenario, it can make or break ganks, in both directions. It's a very good teamwork skill as well, you can help allies escape, or you can speed up your team to help catch runners. Warp-Speed Rammusball is a great way to initiate on your terms, or catch multiple runners, plus it's just funny to see.
His Ultimate can get ridiculous, but it's not great when you first get it. The mana cost is prohibitive, and you can't spam recall yet for perma-rez. Late game you can cast it twice per battle, which is excessive for a cooldown nominally in the two-minute range.
The important thing about playing Zilean, is that he changes what sort of character he is at different phases of the game. Early game he's either an excellent harasser or an excellent laner (because Time Bomb is AoE too). Mid-game he's mostly utility, his bombs do decent damage but his speed/slow means that your team gets out alive and theirs not so much. Late-game he's practically a carry, because his scaling goes into full swing with 2-second recall and 1.1*AP on time bomb.