Showing that CLIQUE can be verified in polynomial time
What you're missing is the difference between decision problems and optimization problems.  "Finding the maximum clique" is an optimization problem.  The corresponding decision problem is "Does there exist a clique of size $\ge k$?".  The latter problem is in NP.  NP is a class of decision problems, so it doesn't make sense to ask whether an optimization problem is in NP; that's only meaningful for decision problems.  Any textbook or standard resource on NP should explain the difference between decision problems vs. optimization problems in more depth.