This process has evolved for me over the past few years...here's how I look at it now:
Step 1. Look at a puzzle/problem.
Step 2. Knowing that there is treasure buried somewhere here (as opposed to real chess games where you can't assume there's a tactic waiting to be found) implies that I need to treat this much more strictly as an analytical exercise with severe time restrictions. In other words, I don't expect to get patted on the back for finding the answer ... rather, I expect to get called an idiot for not seeing it QUICK when I'm told a solution exists. Harsh, but that's part of the training! :)
Step 3. After marking down all forcing lines (checks, captures, threats), start grinding them down via analysis. As a rule, I do the easier forcing lines first as they are less deep and if I hit upon the solution, I'm more time-efficient. You eventually begin to grind the tougher lines until you figure it out/your time expires.
Step 4. Doing it in your head is the goal, I'd rather work on relatively easier problems rather than have to solve tough ones by writing down variations, as it defeats the purpose of pushing the calculation boundaries of your brain at this moment.
Step 5. Referring back to step 2, if I don't see it/solve it within 1-2 minutes, I probably WON'T see it in an actual OTB game, so I stop, look at the solution and figure out if I was just lazy instead of being thorough or was this a genuine visualization/analysis error.
I normally keep repeating steps 1-5 until I get stumped by 3-4 problems and then call it a day. IM David Pruess wrote a fantastic post about this (search these forum posts) in a thread where he indicated that there's only so much new stuff your brain can take in at a time, so forcing stuff into it is usually futile. Setting lofty goals like "I will do 100 problems each day" is just silly. Better to say "I will get stumped and learn 10 new patterns each day".
The key here is to absorb new patterns ... kind of like a few notes of a peculiar musical tune you never heard before. After playing over it once or twice clearly, humming the tune on your own until you get it right, your ear will be able to pick it up even if it is buried amidst a noisy crowded environment or as part of a bigger song. That's exactly how tactics training translates to actual chess game skills.
So in summary, be thorough at solving it and give it your best ... but don't spend too much time either as it defeats the purpose of building your mental pattern database efficiently. For example, if you spend 10 minutes trying to figure one new pattern out as opposed to learning 5 new patterns (with a 2-minute time-limit to figure it out) within the same timeframe.
When solving chess puzzles OTB:
Would it be helpful or harmful to write down variations?
Is looking at the answer on a stumping problem a poor idea?
What is your process while solving chess puzzles?