Preparation
Aside from your Onyxia key, there's little you need to bring to this encounter. Bandages, a couple of mana potions, a couple of healing potions, and the usual buff materials (Elixir of the Mongoose, stamina/agility food, Whipper Root Tubers, Greater Fire Protection Potion) are probably fine. A lot of hunters will bring their fire resist gear with them, but this is unnecessary for this encounter and you're just hurting your DPS as a result.
Positioning
When you first enter Onyxia's Lair, she'll be lying on the ground facing the entrance. Take careful notice of the two mounds in front of her face. This is where you get to stand.
Allow the main tank to enter her lair first and gain aggro. Move in behind them and wait until Onyxia is in position before moving onto the mounds. At this point you will be looking at her tail, the main tank will be looking at her face and the rest of your raid will be split in two other groups standing to the left and right of the main tank. The beauty of this spot for hunters is the fact that they are far out of reach of Onyxia's tail swipe. If you are still learning the encounter, then perhaps the mound trick will help as well. If you are getting hit with tail AoE while standing there, your main tank does not have Onyxia positioned properly. She needs to be moved back towards the wall more.
Phase One
This phase is very easy if you follow a few simple rules. Give the main tank time (we use a minute) to generate some hate, and take periodic DPS breaks every 30 seconds. Without the DPS breaks we've found Onyxia will target someone else on her hate list and fry them. This usually results in her turning to the left or right and mauling the group that's facing that direction. With the breaks, she just doesn't do this. Feign death and drink during the breaks.
Your role is simple here. Wait until the go-ahead is given, then send your pet and start shooting. Onyxia has a tail knockback that will send melee into the whelp eggs. Thankfully, your pets are immune to this knockback. Your pet will take almost no damage during this phase. Don't use any stings during this phase. When you start DPS, use Rapid Fire right away. It should be ready to go again during Phase 2.
At around 68% health, Feign Death and drink to full. You'll need as much mana as possible for Phase 2.
Phase Two
Around 65% health Onyxia will take to the air. At the same time, piles of whelps will spawn in the whelp caves. The whelps will continue to spawn throughout Phase 2.
Onyxia has two main attacks for you to worry about during this phase: deep breath and fireballs. The fireball attack is 1500-2000 damage and shoots out to hit a few people on her hate list. If you're lucky, you'll have a healer take care of this should you get hit, but if not you can certainly keep up with a bandage. If you get really low, pop a healing potion to ensure a 3rd fireball doesn't take you out. The jury's still out on what exactly causes deep breaths, but so far there's 3 factors that seem to be related: you must use as many debuffs as possible, you must spread yourself out and not stand in groups and you must really apply the damage. A deep breath is a 20,000+ damage fire attack that will wipe everyone in the path of it.
Your role during this phase is to ignore the whelps and DPS Onyxia. As soon as Phase 2 starts, get a Serpent Sting on her and keep it on her. Damage wise, go all out and keep your pet on her. Pop rapid fire when it is ready again. Don't feign and drink here, use a mana potion instead. Ensure you aren't clumped up. Around 40% health you'll want to watch the stings. This is a tricky time because if you allow too many dots to drop she may deep breath, but at the same time you don't want dots on her when Phase 3 starts.
Once she is around 38% health, feign death and drink up. She always lands for us between the two mounds you stood on in Phase 1, so try to stay away from that area. The transition from Phase 2 to 3 is a complete aggro wipe, so anyone in the area can aggro her.
Phase Three
Around 35% health Onyxia will land again. As already stated, she'll proximity aggro on whoever is around, so try for it just to be your main tank. Once they've got aggro, allow Onyxia to be pulled back to the same position as Phase 1 and have everyone else assume the same positions as well.
This phase is just like Phase 1 except that periodically Onyxia will shake the ground (fearing everyone) and cause lava to spray up between the cracks. The mounds are an ideal position for hunters, since the fear will not run you into the lava cracks 99% of the time.
Blast her down as fast as possible during this phase. There's a good chance your pet will die half way through Phase 3 here, since the lava damage will eventually be too much.
Summary
Hunters are easily the top damage dealers during this fight. It doesn't matter at all what specialization you are, since your pets are viable for each part of the fight (fire resistance helps on your pet here). We require no healer mana at all for this fight - any damage you take is easily dealt with using bandages.