Chet is one of Pete's best friends, and quite a fixture in U-town. He is intelligent, witty, and very interested in the affairs of the larger world outside the borders of U-town. The last is one of the main things Pete likes about him, he isn't as insular as many of the people Pete knows.
There is one thing about Chet which makes some people a bit uneasy, though. He often seems to be talking to himself.
He's not talking to himself, however, and many people get even more uneasy when they realize this. He's talking to his girlfriend, Randi.
You can't see her (except on special occasions), you can only hear her when she wants you to, but you'd be well advised not to say anything bad about her, because she can see and hear you whenever she likes.
And she knows with great certainty if you've been naughty or nice, and exactly how much of each. She mostly doesn't care, of course, but she knows.
Randi claims to be omniscient and omnipotent, and there's every evidence that she's telling the truth. And, if that wasn't enough to make you take care, she drinks to excess from time to time.
Chet and Randi did not appear in A Sane Woman (though of course Randi may have been there and keeping quiet). Their first appearance in U-town is very near the beginning, as they come into the bar Duffy's, cheerfully arguing about something.
Of course, to those who haven't met Randi, like Vicki Wasserman, Chet is simply a guy talking to himself.
You can read more about Randi here.