With two cards being dealt successively, without replacement, from a 52 card (standard) deck. The probability of being dealt a club SECOND, given you were dealt a diamond FIRST.

How would this be set up? How can I figure this out?

You know that the first card was not a club. There are, therefore, 13 clubs, 13 hearts, 13 spades, and 12 hearts -- 51 cards in total. The chance of being dealt a club under these conditions is

p=\frac {13}{51} = 0.2549