What does Hamlet say where we should read between the lines and his quote has kind of a double meaning?
I think that you should read between the lines with most of what Hamlet says. He often thinks aloud, and seems to think in puns. He is a sensitive and intelligent person with a gift for words, so much of his speeches have double meanings.
One example of something Hamlet says that means something entirely different if you know what he means is when he says to Rosencrantz and Guildenstern "I am but mad north northwest; when the wind is southerly I know a hawk from a handsaw." If you know that Hamlet is just acting crazy then you read between the lines to understand that he saying "I am only crazy when I need to be, but in reality I know exactly what is going on here." Ros and Guil don't know what he is talking about and are left pondering why the wind would matter to Hamlet's sanity and why he would ever confuse birds with tools.
Hamlet uses quite a few puns throughout the play. His first lines "A little more than kin, and less than kind!" and "I am too much i' th' sun" have double meanings, referring to his reluctance to accept Claudius as his stepfather. There are several places, though, in which Hamlet tells other characters that he is putting on an act. In Act 1, Hamlet tells Horatio that he will put an "antic dispoition on," and will pronounce "some doubtful phrase." In Act 3, scene 2, Hamlet tells Rosencrantz and Guildenstern that he is "mad north-north-west. When the wind is southerly I know a hawk from a handsaw." Here Hamlet tells his former friends that he has been deceiving Claudius and his mother, and that much of what he is saying are not words of a madman, but of one who has quite a bit of common sense.
I don't know of a particular instance in the play in which Hamlet tells the readers to read between the lines. Because we know that he truly did see the ghost of his father and we are privy to his plan to act mad, we know that most of his words to those he does not trust are ironic or thinly disguised truths. I hope these examples help.