Quite a few of Moorcock's characters could be seen in that light, from Elric to Jerry Cornelius to Maximilian Pyatnitski, to name only three. Elric is, at most, a reluctant hero, forced into the role through circumstance ... but he as often proves a disruptive element (fitting for a member of a people devoted to Chaos). J.C. -- well, he's someone with an enormous (though fragile) ego, big dreams, and who is really a bit of a fool... ultimately good-hearted, but not at all the mover-and-shaker he sees himself to be, and frequently self-pitying... yet with a saving sense of humor and somehow able to engender exasperated affection in people... a lot like a very bright, often witty, and frequently pain-in-the-neck child..... (Or, as Colin Greenland put it: "The Global Village breeds the Global Village Idiot".)
Pyat is well-meaning, but horrific in his views, self-deceiving, the quitessential "unreliable narrator"; what good he does is more often in spite of rather than by dint of effort; yet he becomes a fascinating character because he is so much our own tendencies in exaggerated form that he becomes very human -- and he sometimes has wonderful dreams (which he inevitably destroys because of his inability to see himself -- or anything else -- clearly).
Karl Edward Wagner's Kane is also an antihero, I'd say -- he is at least as often the "villain" as the "hero" -- and almost consistently driven by ignoble motives... but on occasion by quixotic bursts of nobility or love which usually come close to destroying him emotionally. He is selfish, a murderer many times over, he betrays the trust of those who believe in him time after time, but because he refuses to let himself be used he does often perform heroic deeds that help others -- and yet his arrogance and selfishness earn him the despite of those he helps, driving him even further into himself and away from his links to humanity -- but which he can never quite escape.
Lord Gro, in Eddison's Worm -- a turncoat and a physical coward who performs noble deeds and then betrays those he befriends when they begin to succeed -- driven by a certain feeling for the losing side, whatever the merits of the case.
Even James Bond is something of an antihero. He's a paid assassin; he's often brutal, cold, selfish, self-aggrandizing, and he uses (or has a desire to use) women without any emotional attachment -- yet he finds himself caring more often than he'd like, and it costs him dearly more than once, and he is on occasion willing to be self-sacrificing for someone even at the cost of perhaps failing in his mission. He's also got a romantic streak that trips him up more than once, and which he simultaneously despises and stubbornly sticks to. It's always struck me that, when the series opens, he really is a cold and selfish *******, but he grows to see more of his ties to others until, when he confronts Scaramanga, he recognizes (if not entirely, still to a great degree) himself -- and he's horrified by what he sees.
These are ones that come to mind right off. Perhaps part of the attraction is the more realistic view that gets us away from the older Heroic and Romantic idea that got the stuffing knocked out of it in the last century or so of warfare, when people began to realize that most of that ideal no longer applied, and that the weaponry made it no longer a point of skill or valour or anything but a game of superior weaponry and accretion -- and chance, more often than not; when huge numbers could be wiped out within a few hours -- or entire cities within a few minutes. Thus we began to realize that heroism (in the big sense) was an illusion like so much else. The antihero isn't a true villain, but more like the common person faced with extreme choices... frequently failing through human weakness or greed or fear or simple selfishness, but sometimes showing a streak of compassion and nobility that causes them to do surprising things. They're more complex than heroes and villains in that way, more human. They have doubts and flaws that most people can empathize with -- even while they find them frustrating.
I'd say it's because they're easier to relate to in this age of post-modernism, where the emphasis is still largely on "realism" in fiction, rather than the older, chivalric and romantic models. I've seen it argued that Faustaff is an antihero, for instance... and with considerable basis, I'd say.
A couple more come to mind -- the noir detectives, who are seldom heroic save when forced to be by circumstance -- to fulfill a job they've taken on, or to save their hide. Or even John D. MacDonald's Travis McGee, who sometimes sees himself as the knight in tarnished armor "getting up on that old spavined steed" -- but who'll duck out of most situations if he can... unless prompted by a very quirky sense of personal honor. And even Gollum is something of that sort, as he's occasionally shown as somewhat wistfully longing for the companionship and acceptance of Frodo, and is even gentle and perhaps even caring at times... yet he is hardly heroic in any normal sense of the word -- but he fulfills the quest where Frodo would have failed, albeit by accident ... or chance.
