Yah, this is a tricky one, and probably written to be deliberately obscure because Hamlet (a 30-year-old scholar, remember) is pretending to be nuts. And by the way, Claudius, you may notice, is also confused by Hamlet's answer, so you're in good company.
But, as Polonius later says, though this be madness, yet there is method in it.
So let's clarify the vocab:
A chameleon is the same that it is now--the color-changing lizard. But in Shakespeare's time, it was believed to eat only air.
Dish in this case means the meal, not the thing the meal is served on.
And a capon is a big chicken, over-fed to be nice and fatty for eating.
It's also worth noting that he says this in response to Claudius asking "How fares our cousin Hamlet?"
(Don't be confused by the "cousin" thing...in Elizabethan English, cousin just means relative in general.)
Claudius is asking "how are you doing?," but "fare" also means both food and eat.
So first of all, Hamlet is making a pun on "fare." Claudius ask him how he's doing, and Hamlet answers with how he's eating. This is a bit like responding to "what's up?" with "the sky."
Ok, so Hamlet says what he's eating. He's eating the chameleon's dish--i.e., just air. He then clarifies that the air he's eating is crammed full of promises. And then to make sure Claudius doesn't miss the point about how much nothing he's eating, he adds that you can't get a chicken nice and fat by feeding it air full of promises. So he's now implying that he's starving because of all the air he's eating.
So lastly, how does it relate to his feelings to Uncle C? Well, who's been filling the air with empty promises? Uncle Claudius has. He's been lying left and right and Hamlet knows it: Oh sure, your father died of natural causes. Oh sure, I'm only marrying Gertrude and taking the crown for the stability of the kingdom. Oh sure, Hamlet, I love you like my son and I would never hurt you...
Hamlet, in a nutshell, is calling BS as only an over-educated, angry, young man can. It kinda goes like this:
C: How are you Hamlet?
H: I'm doing great, filling myself up on all this hot air. Try getting a chicken fat on that, and you'll see how I'm doing.
To which Claudius (confusedly? guiltily? what do you think?) says: I don't know what you're talking about.
Hit me up to talk more about Hamlet, Shakespeare, or lit in general.