Vtyjkyuk

So when you're drowning, and you fail your con check, you drop to 0 hp and are unconscious. In the next round, you drop to -1 and are dying, and in the following round, you are dead. It looks like fast healing and Regeneration don't heal this damage.

However, let's say a cleric wades to the edge of the pool and Channels Energy at you after you fall to 0. On the next round, do you fall to -1 anyway? or do you start over and fall to 0 again? Or since you're conscious can you try to swim before you pass out again?

You will be healed, and keep drowning

Being healed above 0 hit points means your character is no longer so close to death, so if she is attacked by something, she has a much lower risk of getting killed due to negative hp. But that doesn't help her against the Suffocation condition. The condition is there because the character cannot breathe, and even if her points are restored, you didn't remove the condition.

To explain the condition in detail:

When the character fails one of these Constitution checks, she begins to suffocate. In the first round, she falls unconscious (0 hit points).

This means that when she failed her first Constitution check, she drops from whatever hit points she had remaining to 0 automatically. And gains the Unconscious condition, which says:

Unconscious creatures are knocked out and helpless. Unconsciousness can result from having negative hit points (but not more than the creature’s Constitution score), or from nonlethal damage in excess of current hit points.

Neither of those two conditions are removed if your hit points are healed back up again. Unconsciousness may happen due to having negative hit points, but there are other effects that can cause it, such as Suffocation and sleep. But let's move on.

In the following round, she drops to –1 hit points and is dying.

Dying is a worse condition than Unconscious and isn't removed by healing hit points either. Because even though healing a creature above 0 hp would make her conscious again, that did not remove the Suffocating condition, thus she remains Dying. In fact, your character could die from extra damage caused to her before she even drowns, if she goes from -1 to her negative con mod from damage.

A dying creature is unconscious and near death. Creatures that have negative hit points and have not stabilized are dying. A dying creature can take no actions. On the character’s next turn, (...). If a dying creature has an amount of negative hit points equal to its Constitution score, it dies.

I removed the part about losing hit points and stabilizing, as it's irrelevant for someone who is suffocating, she might die before that even has a chance to become a problem.

In the third round, she suffocates.

Well, the rules don't spell out what exactly "she suffocates" means, but we do have a clear idea that she is very likely Dead.

Please note that, while Suffocation is not listed as one of the game's conditions it has enough references in the rules (especially spells) to justify it being labeled a "condition" in the game. Keeping in mind that there is no clear definition of what are conditions in the rules, and the core rulebook (and SRD/PRD as consequence) lists them for the GM's convenience (Apendix 2 on the core, p.565). As such, things like poison, diseases and even light is refered to as conditions throughout the rules.

Well after casting Hypercognition on myself, (consulting google) Here's what i could recall.

Drowning

Any character can hold her breath for a number of rounds equal to twice her Constitution score. After this period of time, the character must make a DC 10 Constitution check every round in order to continue holding her breath. Each round, the DC increases by 1.

When the character finally fails her Constitution check, she begins to drown. In the first round, she falls unconscious (0 hp). In the following round, she drops to -1 hit points and is dying. In the third round, she drowns.

Channeling energy

Channeling energy causes a burst that affects all creatures of one type (either undead or living) in a 30-foot radius centered on the cleric. The amount of damage dealt or healed is equal to 1d6 points of damage plus 1d6 points of damage for every two cleric levels beyond 1st (2d6 at 3rd, 3d6 at 5th, and so on). Creatures that take damage from channeled energy receive a Will save to halve the damage. The DC of this save is equal to 10 + 1/2 the cleric’s level + the cleric’s Charisma modifier. Creatures healed by channel energy cannot exceed their maximum hit point total—all excess healing is lost.

As far as R.A.W is concerned, it wouldn't help. You would still have failed your original constitution check. At the start of next round you begin drowning i.e. falling to 0 no matter current hp total.
As a DM, I'd let you try another con check to expel all the water/substance from your lungs. As a player, I think it would be better to use something along the lines of Bears Endurance

P.S. My first ever answer so i hope you like it. Hoped i helped.