Usually the Clifford group is defined to be the group of unitaries that preserve the Pauli group under conjugation, so no proof is needed.
If instead you are asking, how can we prove that a certain unitary (such as the controlled-NOT) is in the Clifford group, the usual straightforward way to do this is just to calculate. Conjugation is a group homomorphism, so it is sufficient to check a generating set of the Pauli group. For instance, single-qubit X and Z operators are enough, so in the 2-qubit case, you should check the action of conjugation for X_1, X_2, Z_1, and Z_2.
See quant-ph/9807006 for more about the Clifford group.