First I look around the board in a blind panic afraid that I'll miss the computer's move; often I do. 
Then I do the briefest of material comparisons; if it looks close I may or may not make it an exact count. Mostly I just want to know if I need mate to survive. Then I look to see to see if anything is hanging on either side. Are there mate threats against me that need to be dealt with? Can I threaten mate? What features of the position might lead themselves to a tactical solution? Does my opponent have pieces lined up for an easy pin or skewer? Is there an overloaded piece somewhere I can take advantage of? Are there back rank issues for either side? The damn pawn structure; can I change it to my advantage? Do I have to change it to prevent his advantage? Calculate like crazy. Look around once more in a panic as the time is slipping away and then "guess and click". Sigh...That's why I'm not much better at tactics than you I suppose. 
What do YOU do when your solving tactics puzzles ?
Learn to move the pieces in your head. You can't improve until you learn that. Once you do, you will improve steadily
I recently decided I'd rather spend 3 minutes, get the question right and lose points than play for my rating.
I think I can digest more looking at less positions in a given section.
I don't touch any of the pieces until I have formulated a solid idea, this could take up to 6 minutes. From there it is usually 15 seconds conducting the sequence, although if it is a really challenging problem it could take another 2-4 minutes.
Of course it doesn't take an average 8 minutes for each tactic, but I would solve the problem in thrice the average time rather than crashing into a blunder 15 seconds into action. I find the process of analysing the board and moving the pieces in my head oddly relaxing and don't mind the potential loss of points at all- especially when the compensation is much greater.
Once you can move the pieces in your head, you have to develop a method for solving tactical problems or otherwise you will be solving at random and your results will be random. I speak from experience. But also I looked at what many strong players did. Not all of them had a method. Quite a few relied on "intuition" which is somewhat random. By that I mean, they sometimes got the right idea, sometimes not, depending on how good their intuition was.
I'd like to know your personal style or system. I need to get my tactics in the 1400 plus range. I know overall I just don't focus enough. I need to slow down and use the time provided. But what do you do? or did you do to get to your level?