First, "chameleon's dish" refers to the belief that the chameleon was able to live on air alone. (See e.g. Nine colourful facts about chameleons.) In fact, there was a proverb that, "A man cannot live on air like a chameleon." According to the Oxford English Dictionary,
From their inanimate appearance, and power of existing for long periods without food, [chameleons] were formerly supposed to live on air. These attributes made the name famous and familiar to many who knew nothing else of the animal. (Quoted by Hedda Haugen Askland.)
Hamlet pretends that Claudius's question, "How fares our cousin Hamlet?" is about Hamlet's food (fare).
Second, the word "air" in "I eat the air" continues the chameleon theme but is also a pun on "heir". According to G. R. Hibbard's edition of Hamlet (in the Oxford Shakespeare series), Hamlet's lines are "a rapid series of pointed quibbles designed to give Claudius the impression that thwarted ambition is the cause of Hamlet's strange behaviour."
Third, "promise-crammed" probably refers back to what Claudius said to Hamlet in Act 1, scene 2:
You are the most immediate to our throne;
And with no less nobility of love
Than that which dearest father bears his son,
Do I impart toward you.
Hamlet may feel this has brought him no advantage. (And in the lines following those quoted above, Claudius had not allowed him to return to Wittenberg.)
Finally, a capon, as Wikipedia points out, "is a cockerel (rooster) that has been castrated to improve the quality of its flesh for food and, in some countries like Spain, fattened by forced feeding." Hamlet says he is not as stupid as a capon: he cannot be satisfied with empty promises in the same way that farmers fatten their capons.
Note that this exchange takes place just before the "mousetrap" play that Hamlet wants to use to confirm the ghost's story about Claudius being the murderer of Hamlet's father. The snarky comments about Claudius's empty promises are therefore meant to deceive the king about Hamlet's true intentions.