According to the Urban dictionary "alligator blood" is a, "Term used in poker.
Used to describe a player who plays fearlessly when short-stacked and wins, or optionally, a player who plays for a long time short-stacked, winning just enough to keep them in the game. Also can be used to mean the complete lack of concern for other people's well being at the poker table." SO,

"Let's play a game of Russian Roulette.
I'll load the gun, you place the bets.
Tell me who will make it out alive."

He is gambling with his live, playing a game, referring to his lives, with an unknown outcome and it's all or nothing. Him loading the gun could possibly be referring to him cheating, as in "loading the dice"

"Over, and over, and over again.
We play the same old games.
Cards kept close to my chest:
That way the secret's kept.
So wear your poker face."

Over and over referring to them playing the game every day, and having to stay alive every day. Cards close to his chest so no-one really knows how he is really feeling, what he is struggling with, and making it seem like the cards he is dealt aren't as bad as they truly are. Wearing a poker face to cover his misfortune.

"Here's to the fool, here's to the thief.
Fiction is bitter and you got a taste."

In tarot cards, "The Fool is always an indicator of newness; as well as the purity and open-hearted energy of a child." And the thief, "in its simplest form will deal with using the mind to analyze the situation and plan next steps to achieve our goals." So, basically he is saying he was innocent, (maybe even naive,) and tried to be open,But was hit hard with reality, and had to over come his problems.

"My alligator blood is starting to show.
I know that you know that I know that you know.
Can't call a bluff with a dead man's hand."

People are starting to see that he is struggling to overcome just enough to stay alive, and he can see that. And he can't call someone else out on their problems, or what they are doing wrong, or try to help because he Himself obviously is barely scraping by.

"Cut me up and wear my skin.
Show me how to live."

They can't tell him how to live, or how to fix what is wrong with him, because they can't understand if they don't live his life, or have his problems and struggles, and it's not as easy as everyone thinks. They don't take him seriously, and would have to walk a mile in his shoes to understand what he is going through.

"Tear me down, clean me up.
Now spill my fucking guts.
Just help me find a way."

He is begging some one to do something, and anything to just him get through, and make him finally be okay.

"Put a gun to my head and paint the walls with my brain.
Put a gun to my head and paint the walls.
~
Now you know.(x3)
Now you know, you can go.
You can give up trying.
Should have just called it quits.(x2)
Leave before it's too late."

He is willing to go to the extreme to end his problems, and now that people can see that his is being pushed to the brink they start to take him seriously. But he thinks they should give up trying help because he feels like it's too far gone to be helped anymore, and to just leave before they are too attached to him when he gets pushed over the edge.

And then the song restarts with, "Let's play a game of Russian Roulette.." Followed by, "So put a gun to my head and paint the walls with my fucking brain." Meaning it's a new day, with the same old battles he has to fight again, and again, but he is still willing to take the risk, and fight despite how close each breaking day pushes him closer to the edge.