There's nothing wrong with Forklift's solution, but here's how to be a bit more methodical. First of all
the possible captures form a directed graph (with a bit of subtlety when pieces' possible paths pass through other pieces that may or may not have been captured by the relevant point):
Now there are some captures we can identify with confidence.
The Ne7 can be captured only by the Nc6, after which the only thing it can do is to capture the Rc8. And the Pa3 can only capture the king. If the Rc8 is captured by the Ne7 then it isn't captured by the Bg4, whose only other available capture is the Bf3; it then has nothing to do but capture the Nc6. What captures the Bg4? The only thing that can is the Ne5, so we know how everything begins: so we have the sequence Ne5-Bg4-Bf3-Nc6-Ne7-Rc8. The Rc8 can now capture the Bh8 and nothing else can, and then the Bh8 has nothing to do but capture the Ra1. So Ne5-Bg4-Bf3-Nc6-Ne7-Rc8-Bh8-Ra1.
it's obvious that the only way to finish is Ra1-Bc1-Pa3-Kb4.