Sibling was always better at everything (whether that's true, or just the character's perception doesn't really matter), leaving the PC with a sense of never having been 'good enough.'
Doesn't have to be a full sibling, could be one of those situations where a child was adopted by a family that didn't think they could have a child of their own, and then had a child later, leading to the adopted child feeling redundant and unwanted.
Or the older sibling passed away, and regardless of the exactly wonderfulness of the dead sibling, the survivor felt like s/he could never measure up to the rosy-colored memories of the wunderkind.
For the class, a fighter could be based around more underhanded tactics, such as trip, disarm or sunder maneuvers, taking the advantage away from the opponent.
A rogue would automatically suit the envy theme, being a class focused around taking things from others (things that maybe they feel the other didn't earn or deserve?), and whose combat techniques are based around using deception and guile to overcome those who may be physically more powerful or martially skilled than they are.
An abjurer could focus on counterspelling (which I wouldn't advise, since it's a terribly option, for the most part). Most types of specialist wizard would be tweakable in that direction, really. Someone who is resentful of the abilities of others might use enchantment spells to turn the boastworthy abilities of rivals to their own service, or conjure up creatures from other planes to 'even the odds,' or transmute themself (or their foes), or use necromancy to drain the strength and abilities of others, etc.
A ranger's favored enemy could represent the sort of invidual that the character is specifically envious of, taking out their frustrations on whoever it is in their past that they can never equal up to or prove themselves to.
RP flavor things could be to channel those frustrations into their combats. At the end of a particularly grueling fight, against a 'big bad,' the PC might inadvertantly say something like, "You always thought you were better." indicating that they aren't just seeing the big bad evil dude fall, but also channeling some of their frustration towards whomever in their past made them feel envious.
@Set: I like the sibling envy idea, namely the dead sibling. The PC's inability to truly get rid of these feelings cause him to misdirect it (towards enemies). I like the fighter idea and the ranger idea a lot. The rogue would use Feint primarily, in this case. The transmuter idea is what REALLY appeals to me, much more so than the abjurer. I know that in Thassilon transmutation was greed and abjuration was envy, but I think they should really be the other way around.
@Gallard Stormeye: An enchanter could definitely work, but it doesn't quite fit the theme. That would be closer to lust, I imagine than envy. The Demon or Fey bloodline sorcerer could definitely work though.
Along with the suggestions above: Be fundamentally unhappy with yourself as you are, the only relief for which is to take on those aspects of others you feel would complete you. And like a drug, it only helps ease the pain for a while.
About envy towards the party that won't result in grief for the group:
Try too hard. Be a "poseur". Keep reaching for that special something the beautiful, the talented, and the loved have to fill that hole in yourself that can never be filled.
People seem to look up to the party paladin quite a bit. What is it about him that does that? It has to be that almost absolute certainty he has in himself. Yeah, that's[ it. Emulate that above all else!
People listen to the wizard. Always has something smart to say. Reads a lot. Spouting off what you've read elsewhere is certainly enough to look smart, right?
People adore the bard. But you don't have any musical talent.... But all you really need is an instrument at your side and people will start paying attention, right?
Admittedly this would probably result in a character more pathetic and sympathetic than you may intend, but it could easily go into creepy Single White Female territory if played right. ;) Or wrong, depending on the group...
Also, on the matter of envy itself, has your character opened his/her heart to Zon-Kuthon?
Bard might be a good choice. Back when I was a semi-professional musician, there was a lot of envy around. Envy over who wrote better songs, who played guitar faster or flashier than who, who had the better voice, who got chosen to be the drummer, etc.
Guys were like lady gossips in a knitting circle with how bad they ragged on each other. Lots of backstabbing, lots of talking behind people's backs.
A bard knows the name of every musician, poet, writer and actor in the kingdom, and has opinions on precisely all of them.
I like the idea of an ugly, innocuous style enchanter who is actually competent, but feels they have been overlooked and ignored by the wider masses. They crave attention and adoration but never receive it – mostly because they’re plain, selfish and inwardly unhappy.
This character hate, hate, HATES it when the spotlight is on a rival hero that they feel is unworthy of the people’s love. A paladin slays a ferocious dire wolf outside the village and gets a feast in his honour? Everyone loves him!? This enchanter will concoct a plan to see the paladin magically enchanted to kick a child and then revel (alone) in the poor paladin’s destroyed reputation.
If I'm not loved by all, nobody can be.
I'm going to have to echo the Inquisitor sentiment - particularly playing up the aspect of no one being faithful enough, purely due to the doubts clouding their own heart. The twist, of course, is that they are loyal BECAUSE of their doubts, which makes them as effective as they are at rooting out corruption among the faithful. And while this may make the inquisitor a fairly disliked individual (due to revealing the ugly facets of faith and the conflicts within), that would not mean they were not respected for doing the job that they do.
They envy the faith of others, and seek to disprove it at every turn, if only to feel more grounded in their own.