This problem is clearly NP-hard since it is a generalization of many NP-hard problems. The most obvious one is the maximum clique problem, i.e. finding a clique of the maximum size in a graph. The decision version of this problem is a special case of your problem where you set $t$ equal to one. (See below for the hardness-proof).
Other example is the triangle packing problem, assuming you are looking for pairwise vertex-disjoint cliques (note that for the maximum clique it does not matter if the cliques have to be vertex disjoint since we chose $t=1$). In the triangle packing problem we are looking for a set of $k'$ vertex-disjoint triangles ($K_3$) in a graph, which is a special case of your problem where we set $k = 3$ and $t = k'$.
Note that the maximum clique problem is one of the Karp's 21 hard problem, which are probably some of the most important/original proven hard problems. In his paper "Reducibility Among Combinatorial Problems" [Karp, 1972], he provides some of the very first hardness reductions using Cook's theorem. Here is a link to the wiki page of the paper where you can find more information and a link to the original paper.