Powerset construction is used to convert a non-deterministic finite automaton (NFA) into a deterministic finite automaton (DFA). Is the method/algorithm used to do this deterministic itself and if so, why?
The method itself is (or at least can be made), though the question is somewhat ill posed. What do you mean by deterministic?
The algorithm cannot be run on an NFA or DFA, so it's not deterministic/non-deterministic in that sense.
In most cases, the algorithm will run on a Turing Machine, or some similar model (RAM machine, programming language, lambda calculus, etc.).
It's known that any non-deterministic Turing Machine can be simulated by a deterministic one, using backtracking. So in this sense, the algorithm certainly can be made deterministic.
Is it possible to define the powerset construction in such as way that it's non-deterministic? Yes. There's some element of choice in which states you process first, which NFA states you choose to expand, whether you look breadth-first or depth-first, etc.
But it's trivial to remove these choices, either by resolving them arbitrarily, or by making the choice with some heuristic.
The usual description of the powerset construction corresponds to a deterministic algorithm whose running time is polynomial in the output size.
Although non-deterministic Turing machines are equal in power to deterministic ones, they are (probably) not equivalent in terms of complexity (a particular case is the well known P vs. NP conjecture).