@ sunbreath; Cool name
Ahem, fighting Eldar. Well, it's best if we do it on a Force Org structure.
How to kill Farseer; He will most likely be on a jetbike, so he has MeQ statline (although he is insta-gibbed by T6), and his normal 4+ invul. There are three units ideally suited to fragging such a unit; Vespid, a Centurion Shas'O and an Ionhead. A 9-strong squad of Vespid, with a boost to BS4, will take apart a Farseer (2.5 unsaved wounds). Likewise, a Centurion (PR+CIB) will deal 2.4 unsaved wounds, and if the Farseer fails to save against the PR, insta-gib ensues. Thirdly, an Ionhead is a perfect weapon (especially if you can get in range to use the BC's). With the Ion cannon alone you can slay him, because he's only likely to refract the first S7 hit off his 4+ save, the other will get through and insta-gib. Alternatively, a FoF will deal 3.75 unsaved wounds, which is enough to wipe him out.
How to kill Autarch: Again, he is likely to be on jetbike. Some people take him with the Warp Spider jump pack, but it's quite random and has less buffs than a jetbike. Same principles as with Farseer; hit him with massed AP3/2, or an Ionhead. Failing that, FoF the dude.
How to kill Avatar: Kill the rest of the Eldar first. It's slow, short-ranged and makes nearby units Fearless. If said units are dead, it's no longer any use. When you're done killing everything else, hit him with Ionheads, MP's and PR. If he threatens to melta something or charge, tie him up with Kroot or Gun Drones.
The characters are not commonly encountered, Eldrad+Avatar is common (but pretty useless, for the same reasons that the Avatar is useless). The other characters are either slow, or chained to joining an Aspect squad of the same type. Seeing as you'll face Aspect squads anyway, they give the same buffs+Fearless, so the Aspect characters are essentially Exarchs on steroids.
How to kill Scorpions: Mechanise. Static lists are going to especially vulnerable to these guys in 5th Ed (because of the changes to Infiltrate), but if you have everything moving 12" and firing, Scorpions lose out. Lacking Fleet, they can't catch you. A FoF will really put the hurt on these guys (3.75 casualties), and using Firestorm teams and submunition rounds you can clean them up. A Vespid squad is really nasty as well, able to drop 5 Scorpions at BS4. Centurion is also highly effective.
How to kill Fire Dragons: These guys are the death of vehicles and Crisis, but useless in combat and at shooting infantry (unless they get the flamer on the Exarch). A good policy is to hit them with a FoF, a Firestorm team or Vespid, then charge Kroot in. The Kroot will win combat, and you can finish off the retreating survivors at your lesiure.
How to kill Wraithguard: Well, much like with Avatar, I wouldn't bother until you've killed everything else. You'll face one of two combo's; large foot-slogging squads with a Warlock with Conceal (giving them 3+/5+), or 5 x WG and the Warlock in a Serpent. For foot-sloggers, ignore them. For the Serpent squad, disable the transport, then ignore them. If either threatens to get in range, tie them up with Kroot or Gun Drones.
How to kill Banshees: You'll usually face a mechanised squad of these girls, due to their relative fragility. Disable the transport, then drop submunition rounds on them (using markerlights to reduce any cover saves they may have). Firestorms or a FoF will also kill them, and charging in Kroot wastes their power weapons. They can Fleet, and on the charge they are nasty in combat, so once they are foot-slogging keep your distance and hammer them with fire. A good FoF will also clean them up quite well, given their mediocre 4+ save.
How to kill Harlies: These guys will definitely be in a Holo-Falcon. With the reduction in speed for Eldar (due to 5th Ed nerf of skimmers), disabling their transport is made a little easier. Once they are out, hit them with Firestorms and FoF. Massed S5 will really take them apart, due to their 5+ save. You can potentially wipe out an entire Falcon squad with a single FoF. Much like with Banshees, stall with Kroot and Gun Drones, and hit them with ranged firepower.
How to kill Dire Avengers: These guys try to beat us at our own game; their Serpent of Fury pumps out more bolter shots at slightly greater range, but they have to wait to reload next turn (should they choose to Bladestorm). Disable the transport (seeing a common theme yet?
), then drop submunition rounds into them. They have a move+shoot range of 24", so be careful with Crisis, only move in to mop up the squad. Exarch can be effective in combat, but generally he'll pack dual catapults and Bladestorm. A counter-FoF is quite effective, as the Wave Serpent is unlikely to kill the Devilfish (unless it takes EML/BL, and even then it's 1 shot, which only damages on a 4+, and you have a 4+ save from your DP), and your pulse rifles will frag more Avengers than they will back.
How to kill Rangers; Two options here. Firstly, use Kroot to counter-Infiltrate, then hold them up in combat. Secondly, use the AFP. It wounds them on 3+ and ignores their spiffy 2+ cover save, and pins to boot (plus its large blast). Combined with their average Ld and medium squad size, you have a reasonable expectation to wipe them out, or pin the survivors.
How to kill Guardians: You really only see them in themed armies. Against Tau, Firestorms, submunition rounds, FoF, even Kroot can pretty easily beat these guys. Medium-large squads, trying to protect a heavy weapon. While they do it better than IG, it's not terribly suited to the Eldar way of war; fast and specialised. If there is a Warlock giving them cover saves, drop AFP onto them.
How to kill Shining Spears; These guys are basically the bane of Crisis and skimmers. However, they are only a small squad, and fairly expensive. A Vespid squad will decimate these guys, especially when backed up by markerlights or a Centurion. If you don't kill the last squad member, charge into combat to deny them movement and shooting (you might even finish the last guy off, their awesome S6 power weapons only activate if they charge, same for the S8 power weapon on the Exarch). Without the charge they are pretty weak in combat.
How to kill Warp Spiders: Very annoying unit, very fast as well. They are the death of Crisis and infantry, and they can also glance skimmers rear armour to death quite readily. Reasonably priced, and a staple of many Eldar armies. Use something to lure them out, then hit them with your Centurion and Vespid. Assault any survivors. Failing that, hit them with a FoF, Firestorms or submunition rounds.
How to kill Swooping Hawks; These can be annoying with their grenade pack ability, but generally they are not very effective against infantry. However, they can be extremely lethal to skimmers, hitting on 4+ with haywire grenades is really effective. Submunition rounds, failing that hit them with massed S5. Their lasblasters are essentially hellguns, so outside of combat they are not very dangerous. In combat with infantry they will also die; Vespid shooting+assault is quite good as a counter, considering Vespid have the speed to catch them.
How to kill Vypers: Ion cannon, MP. Deathrains and Firestorms are very effective at destroying these fragile skirmishers. Massed S5 can also kill them.
How to kill Support weapons: Submunition them. In most cases (aside from vibro-cannon), their firepower is short-ranged, so shooting them up from long range is best.
How to kill Dark Reapers: Their AP3 is death to Crisis and infantry alike, so prevent them from shooting. They'll usually be camped with a Farseer at the back, raining death into your lines. Vespid are fast enough (and can move through cover pretty quick) to get to them and cause havok. Shoot them up, then assault. In combat you'll either draw or win, because they will have less bodies and less attacks. Even if you lose the Vespid, they'll lose shooting time, which gives you the oppertunity to move up your vulnerable Crisis and infantry. If you can't shift them, a FoF (they can't hurt AV12+) and submunition rounds will whittle away at the squad. Take out the Farseer if he is nearby (he'll usually be attached though, so just shoot the squad).
How to kill the Wraithlord: Despite the slowness you'll wanna kill him, because his brightlance will destroy vehicles. However, he is quite tough, so you are usually better off tying him up with Kroot. While he mashes 7pt infantry up slowly, you can move your vehicles around safely. Don't hestiate to charge, he has two flamers and will obliterate Kroot if they hang back.
How to kill the Falcon; Arguably the toughest vehicle in 40k, 5th ed rightly nerfs it. It is still highly surviable, so preventing movement and shooting is key. MP, ion cannon, seeker missiles...and of course Piranha fusion blaster and railgun helps with disabling it. Try to outflank it; lure it's passengers (Harlies, Banshees, Wraithguard) to a target, then send a unit over/around it to get at the AV10 rear.
How to kill Fire Prism: Disabling this vehicle is of greater import, as it's a powerhouse of anti-everything shooting. Two Fire Prims+Harlie Falcon is a tough and effective setup. Combining shots gives them a single twin-linked railgun blast, or a twin-linked AP3 submunition blast. Use MP and ion cannon to stun/shake, finish them off with fusion/seekers/railgun.
Strike hard, strike fast.