depends on how you define 'melee DPS' since most of the best 'competitive' characters have some kind of ranged attack.

I personally would suggest kassadin He may be an AP character, but his basic attack 'is' a melee one. And his ult is amazing. If you're looking for a non-ap character (a 'non-caster' per se), I would say Tristana. She is, amazing.

How is Nasus a top tier melee dps? He is incredibly item and farm dependent and can't gank for his life at the start. I understand this is what constitutes a low tier champion.

I would agree with choosing nidalee, she's a good melee dps in cougar form and also is able to be a ranged support.

So you basically get an evened out character who is able to be both support and dps quickly.

nasus is a tank with added utility and pushing ability, and he adds alot to group fights with his ult. He's a decent character in case your whole team all wants to be insane glass cannons: Nasus can push the lanes, throw the wither, and tank in support. He probably won't result in the best score, but he will definatly help you win.

Of course, I'm only mid ELO so maybe high ELO games 'are' all glass cannons. I was under the impression that most high ELO groups want at least one tank.

He's definatly not a huge DPS tho.

Nidalee.

Shaco: oracles and win
Jax: SotD and win
Nasus: EC and win (well, not entirely, but thats the basic idea)
Pantheon: Perhaps up there..
Yi: Lol? Srsly?

are you sure about that?... none of these champs mentioned are viable in high tier play...
jax<sword of the divine
shaco's just kill 1 person strong
trynd's horrible now, no one plays him... (in high tier)
yi's being shat on as well, warwick's always been probably the weakest link in ANY game...
pantheon i still have my doubts about
nasus is a good champ, and can be built dps, but i don't see anyone doing it. he's definitely viable tho.
the only fairly decent suggestion was nidalee and even she's not that high tiered.

if you're looking for carry, then may i suggest corki, as it stands he's one of the best dps right now.
katarina's pretty tank too but with the recent reworks i'm not so sure. Tristana's... well, dec. all i gotta say. twitch has always been a high damage champ... that's all i can think of around the top of my head right now.

Viable options being, Gangplank (not as much due to nerf on magic parleyy) Shaco, and Nidalee which is the most played in High elo at the moment.