When you go to the dogs, you become ruined. You might say, for example, that if your friend spends all his hard-earned money on lottery tickets, he's more likely to go to the dogs than to win a million dollars.
The saying go to the dogs has been around for a very long time. It seems to have started in the sixteenth century from the idea that useless or spoiled things, like food, should be thrown to the dogs rather than be eaten by people. Rotten or ruined things are suited only to go to the dogs; a person who ruins or destroys his life, therefore, goes to the dogs as well.