I am looking to run a campaign of evil characters, but I want them to have consequences for being evil. The more evil power you gain the more it changes your physical body, mental state and ability to deal with normals. Are there any specific RPG's that handle this?
If you're looking for a game about the consequences of evil, check out Kill Puppies For Satan. It completely deflates the idea of evil as a grandiose, melodramatic pursuit and shows those people who barter for dark powers as being cheap and a little pathetic. All the consequences you mention in your post are present.
Ravenloft for (A)DnD is an excellent choice for the exploration of evil roles in a gothic horror / fantasy setting.
What you get with it is a rules expansion for (A)DnD that allows you to measure - in a pretty unobtrusive way - game effects like fear and horror (that the characters experience), and that also presents you with a multi-level, gradual system to track and measure the evildoer's descent into madness, inhumanity and corresponding, symbolic, physical mutation. This systems also presents incentives for the evildoer, trying to "lure" the character deeper and deeper with rewards that partially compensate them for their degeneration.
As for the world of the setting: Officially it consists of "domains" that map 1) the most famous literary pieces of the gothic horror genre and 2) samples of DnD's most famous settings into a relatively(!) consistent whole. However, due to the nature of this setting - the "domains" being separated by "The Mists", a secret, godlike power(?) -, it is extremely easy to bring anything of your own design under Ravenloft's rules.
Personally, I'd recommend the version published for DnD3.5, but most other editions are great as well. (In fact, the earlier the books, the better they look. Poor 3.5 version is full of pretty weak illustrations that can be a great turn off for a certain kind of people... like me, for example. I do own the previous editions though, and can fall back to those when in need of visual inspiration, relying on 3.5 for rules only.)
Note, please, that Ravenloft, by default, does not encourage playing evil characters. PCs are supposed to be heroes tempted by evil and resisting it to the best of their abilities. However, the rules hinted at above provide a solid yet simple system that could meet the requirements of the Q.
PS: In case you're not into (A)DnD, you might want to take a look at Kult, which, though an old game (that definitely belongs to the horror genre, so reader discretion is advised, again), also has a rather efficient system for the measurement of characters' mental balance (+/- extremes) and corresponding mental/physical changes. Kult, by default, is set in a relatively contemporary setting - however, it features dream realms in which a whole isolated fantasy campaing could easily be set up.
No-one has mentioned My Life With Master yet?
You play evil minions of an evil master who asks you to do despicable things. The more evil things you do the more you hate yourself (the self loathing stat goes up) and the less you're able to overcome your fear of the master and rebel to him. However, you can discover that you love someone and refuse to obey, making your rebellion more effective.
Wikipedia's My Life With Master article has a more detailed description of the specifics of the game.
I once played in a game based on Glen Cook's Black Company novels and that worked really well. We played with the themes of honour amongst thieves, loyalty, betrayal, the lesser of two evils and so on - very epic in it's own dark and twisted way.
Green Ronin released a d20 supplement based on the series, but I never got to see how it worked in practice as our GM considered d20 excessively complex. I don't know how much it played on the themes of evil we saw in the books, but it might still be worth trying to pick up a copy if you like d20.
From a more modern perspective, you may want to have a look at Black Crusade.
With the option of playing campaigns at both Dark Heresy level and at the Deathwatch level, this could make for a very interesting campaign, if you like the Warhammer 40k universe. It is a pretty dark setting anyway, without bringing the Dark Powers into the heart and souls of the PCs. I'm certainly looking forward to playing this at some point.
I'd say that a Fate based system like that in The Dresden Files RPG would work extremely well and could be converted from present day Earth to any setting you want effortlessly.
I say that it would be a great foundation because the system is very much about building a person that happens to be a little (ok, a lot) more than average, with a sort of power/free-will tradeoff. You can spend "refresh" to buy powers and mortal stunts, but unspent refresh gives you more fate points. These fate points allow you to influence the world, with declarations and invocations of aspects, or buy out of compels and ignore your nature (e.g. spend a fate point to not eat the king's face right now like your nature says you should want to). You could also accept the compel and gain a fate point for giving into your nature.
Characters have aspects that help define their character as a person. For a simple example, a dark angry batman type, pretty far into the not-yet-capital-E evil spectrum, might be something like this:
In any sort of social (or otherwise) situation, the GM can offer a fate point to the player and point out that one of his aspects might be making him want to... do something. The player can accept the fate point and agree that they do that thing, or spend a fate point to ignore the compel (a player could also self-compel, with GM approval). That fate point can be used for anything from re-rolling dice, to getting a bonus to a roll, or to declaring something into existence fitting for the environment/situation (subject to GM approval/acceptance).
By default, your character has a set number of stress boxes that work a bit like temp-hp in 3.5/PF for social/mental/physical stress, determined by their stats, along with the ability to take consequences if they run out of stress boxes. Consequences can be anything, and are essentially just another aspect fitting to the "attack" that spawned them, which the GM can compel against (by offering fate points - the player can ignore these by buying off the compel). Consequences go away over time at various intervals based on their severity: mild (2 stress), moderate (4 stress), major (6 stress) or extreme (8 stress). They can also be used, with GM approval, to do things like taking a consequence in exchange for improving a roll - sort of like how some games let you use your health as mana in a more extreme and character affecting way. For a blunt and simple example, a werewolf might take a bonus on an attack in exchange for a "clouded with rage" consequence.
Taking consequences is also the result of physical/mental/social combat, the consequence fitting with the sort of character you have and whatever caused you to take the consequence. This can give you the consequences of "being evil" and whatever it is that makes you evil, which can then be used as compels (which sounds terrible, but work great in practice once everyone gets the hang of things) to have a real impact on things, as opposed to D&D/PF's "I'm a barbarian and talky people are talking so I'll sit here staring at a wall while talky people talk".
Personally I would recommend Kobolds Ate My Baby for an evil campaign, but that's probably not what you are looking for.
D&D3.5 supplements Heroes of Horror and Book of Vile Darkness have variant rules such as "Taint of Evil" and "Lingering effects of Evil".
Black Crusade has mechanics for corruption and insanity.
Characters in Vampire: The Masquerade have a humanity stat and consequences for loosing points from it, but it's a bit sketchy for your purposes.