Is it possible to encode an arbitrary computation as a series of NP complete problem instances?
Each program instance can be encoded as a (possibly very long; even infinite, if the program doesn't halt) series of NP complete problem instances, e.g. as follows:

For each "wire" inside a computer, we can derive a SAT formula that describes how the value (on or off) on this wire after the following clock signal depends on the currrent values of several other wires.  Using such SAT instances along with a set of variables that describe the current state of the computer, we are able to simulate any computation through a series of SAT instances.

Of course it is possible to combine multiple of these instances (or use a different approach), in order to simulate larger chunks of the computation at once. But there are limits:

If your original problem is in NP, then of course you can encode it as a single NP complete problem instance. But you probably will not gain much from doing this.

If your original problem is not in NP then you will need a superpolynomial number of NP complete problem instances in order to solve it, since a polynomial number of NP complete problem instances can be solved in polynomial time by an NDTM, implying that the problem is in NP.

So the short answer to your question is: "Yes, but usually not in a way that is helpful in the sense of your motivation."