Sivir - With her ricochet she just eats teams alive. In my opinion the best ranged carry in the game.
Ashe - Anyone who has seen an ELO late game Ashe knows why you get her first. End game her non-crit hits do 450 or more.
Corki - Moved him up because after his "buff", he has become a significantly more effective carry.
Pantheon - After his buff you just cannot ignore him anymore.
Anivia - The more you kill her, the harder time she has of getting ahead for the rest of the match.
Akali - Surprise? Not even. She takes your carries out so fast. It is the job of every Offtank to get rid of the anti-carries.
Miss Fortune - Not as good as before, but still a ranged carry.
Teemo - His blind and poison can faceroll any champion, especially your carry. The faster he dies, the faster you destroy their damage output.
Kennen - No one likes a painful team stun, so the sooner hes gone, the sooner you have an advantage. It also helps to get close to him because his main damage is poking from afar.
Kog'Maw - A few weeks ago I would have put him at top, but he lost some of his effectiveness lately. He is still dangerous, and therefore is still in Top Priority.
Morgana - If she gets her ult off, you are screwed. Don't let that happen.
Sona - One of the best AoE stun / Support champs in the game, and can keep her team alive just enough to kill your entire team. She needs to die.
Fiddlesticks - So I never really thought of Fiddle as much of a threat, but that Ult of his is basically guaranteed a kill if left unchecked.
Ryze - He has some serious nuking power... Try to eat him before he starts his combo.
Twitch - Sucks now, has way less damage. Kill him just to get rid of him lol.
Tristana - She would be in the Top Priority tier, but because of her amazing escape skill, she shouldn't be bothered with if you can take someone out from the Top Priority tier first.
Annie - Nobody likes an AoE nuke and stun, and she goes down super easy anyways. Just get rid of her before she makes a scene.
Twisted Fate - He has that annoying stun, and can escape fairly well. Depends on if he is their main damage or not, but he's in High Priority.
Veigar - Eats your casters. Can't let that happen. Must die.
Nunu - This may seem like a strange place for him, but you can easily destroy his usefulness. His ultimate "charges" as it channels, so the earlier you stun him the earlier you ruin his ult, and the less damage he does. Good Nunu players go AP, which is why hes higher. Tank Nunu is obviously useless, and shouldn't be targeted.
Ezreal - Hard as hell to catch, but squishy once you do. I found that is actually a lot easier to get to him with Frozen Mallet, so I moved him up. He is still a carry, and must be stopped.
Shaco - Once he realizes you aren't scared of him, he will **** his tights and Deceive the **** outta there.
Katarina - Her ult is devastating, and her pokes hurt. But her damage output is decreased when you break her ultimate. Just poke her and you are done.
Xin Zhao - He has some really annoying CC, and can be dangerous if left unchecked.
Lux - She gives the illusion that she's hard to catch, but once you do, she's an easy kill. Her CC is a major nuisance.
Gangplank - His crits end game can 2 hit your casters, and that is bad. Very bad.
Warwick - Not a huge priority, but his ult can really screw up your team structure.
Janna - Support that needs to die, but has really good escape methods. Not worth chasing over the high and top priority tiers.
Taric - Taking him out of the fight is necessary. The only reason he isn't higher is because he is bulky... ish. depending on how he's built.
Vladimir - You will want to make him waste his pool so when your team comes in after you he has no real escape.
Master Yi - He's annoying. 'nuff said.
Evelynn - Her stun is surprisingly annoying, and the sooner you get rid of that, the sooner your teammates can stop raging. I moved her up to Mid Priority because i find that good Evelynn players are far more dangerous than i originally gave them credit for.
Soraka - Just because she is support.
Malzahar - Meh. Easy to avoid, until he ults. THEN you want to get rid of him.
Kayle - Not a hard carry, but she can do some damage with her slow. The only reason she's lower is that if you focus her, she will just ult and make your efforts useless.
Heimerdinger - He should die, but hes not your problem. He does damage whether he is disabled or not.
Kassadin - He does some heavy damage, so scare him off. He's a ***** to catch though, so don't chase if he runs off. I moved him down a tier because of his Ult. Attack him just enough to scare him off, but DO NOT CHASE.
Karthus - Hes a huge pain, but he keeps on chuggin even after you kill him, which is almost more dangerous because he has nothing to be afraid of.
Nidalee - Also needs to die, but amazing at kiting; something you cannot deal with.
Swain - Stupid health regen, and lots of annoying CC. His damage output is a little low, so he isnt a higher priority.
Sion - Going for him can waste his stun on you, which is good, but he is tanky as hell.
Gragas - One of those bulky champs that should die but just isn't the best for anti-carry focus.
Mordekaiser - Mordekaiser has impressed me of late. With his recent HUGE buff, he is far more dangerous and actually changes the tide of the battle even without his Ult. Because of this buff, however, people don't build him quite as tanky as before, so between Madred's Bloodrazor and Frozen Mallet, he may be someone you want to go for.
Udyr - He's a serious pain for your team, but mirrors are always really risky, because if he has slightly better runes [assuming you dont have my recommended runes] or he is slightly farther along on his item build, you are screwed. He is also really bulky, so i suggest saving him for later.
Just a little side note for this part of the priority section, I actually stared at the remaining champs for about ten minutes trying to figure out how to go about this. I'm not even going to write a reason to go for them, because you shouldn't. I will explain why not to. Most reasons will be "tank blah blah blah" so don't get excited.
Poppy - She is an anti-carry, but her ult makes it useless to target her.
Cho'Gath - Lots of health scales well with your madred's, but most of them go tanky, and so it takes forever to kill him in a team fight. Time that is wasted.
Zilean - If you focus him, he has to blow his ult on himself, which is good for your team. But like i said. He has his ult. Meaning he will just revive.
Olaf - His ridiculous lifesteal is just dumb, and he has burst, which eats you up. Try to avoid.
Dr. Mundo - Another one of those annoying ults that you just can't seem to handle in a teamfight. Ignore him if possible.
Blitzcrank - Lots of annoying CC, and usually pretty bulky. His damage is less than stellar though, so don't bother in teamfight scenarios.
Urgot - His poke hurts like a *****, but there's no point in targetting him. Way to bulky. I moved him a little farther down because I find his shield and natural bulk super annoying.
Nasus - You really shouldn't ever 1v1 him with his ult and natural lifesteal. Just a bad idea.
Rammus - If you can get rid of him, that means no AoE ult and no taunt. But he usually stacks anti-AA stuff, including his Defensive Ball Curl.
Amumu - That % health damage from Despair hurts like a *****. Sadly, he's a tank, and he wants to get hit for Tantrum.
Shen - There's that taunt again. Unfortunately, he's an incredible tank.
Malphite - If you get in melee range, he can use ground pound. It hurts. A lot.
Garen - Sunfire Cape plus judgement is very bad for Udyr's health.
Galio - With Bulwark, he likes taking the hits, and it makes his Ult that much easier to activate.
Singed - The only way he is effective is if you chase him.
Alistar - All he will do is stun you and then push you. Waste of time.
Tryndamere - Okay. Maybe it is important to focus him, but it isn't your job as Udyr to focus him. Anti-Tryndamere champs include ranged carries, AP carries, CC champs, and burst champs. Udyr is none of these, and while he has a stun, that is only once every six seconds.