This is a decent question because it tries to delve into the thinking process of solving a tactical puzzle. Of course, Tactical motifs and Positional motifs are important but you gain those by experience or study time, and I believe what the asker is looking for is a methodology he can apply to a difficult puzzle in hopes of solving it. So, with that in mind:
Note all captures, checks, and unprotected or insufficiently
protected pieces. Visualize each separately, alone. When you do this, you begin to connect some dots in the position, like "if I take here and he takes, that leaves this piece undefended...", etc., etc.
Note all potential double-attacks, probably the most common tactic in chess.
Visualize all avenues of attack for every long-range piece - Bishops, Rooks, and Queens; this means mentally ignore any obstructing enemy or friendly pieces, i.e., Bd3 looks down the b1-h7 diagonal regardless if there are black or white pieces in the way. This paves the way for clearance sacrifice ideas, and discovered attacks.
Visualize all potential knight-forks, and all knight-forks after the move of the knight.
Honestly, I would start there and avoid using the computer. It will not teach you anything you cannot figure out yourself - you just need to be persistent.
I would also write it all down so you can drop the puzzle and come back to it later. You are gaining good habits by doing this, and it gives you valuable references.
Only then when you are completely spent on the puzzle do you pass it to Stockfish.