Showing a problem is NP complete? Reducing CLIQUE to KITE.
With regards to the reduction, you need to show that given an oracle for Kite, you can solve Clique in polynomial time. It doesn't really help that every Kite has a Clique in it: we're interested in solving Clique, but a graph that has a Clique doesn't necessarily have a Kite (if it has a Kite, then it certainly has a Clique - but that's not enough). You need to modify the graph so that if(f) it has a Clique, then it also has a Kite and pass that modified graph as an input to the oracle.

We don't ordinarily believe that an NP-complete problem can be solved in polynomial time. The point of a verifier/certifiate is that with help, we can solve it in polynomial time. The certificate is the "help", and you can define the certificate to be almost anything you like. For instance, for Clique, we can ask that the certificate be a list of the vertices that make up the Clique. Verifying then becomes easy: we can check we've indeed been given $k$ vertices, and that there are edges between all of these vertices. If you like, you can think of the certificate as a solution to the problem, and the verifier's job is to check that the solution is correct.