For me the easiest things are usually also the cheapest things:
0. Check fuses
1. Make sure you have fuel in the tank
2. When you turn the ignition to On you should be able to hear the first DME relay under the drivers seat click
3. When you start to crank you should hear the second stage click
4. Check for spark (old spark plug connected to one lead grounded to the engine metal while someone cranks). If you have spark you can assume a good DME relay, and a DME that hasn't catastrophically failed.
Next, you need to make sure to get fuel. Again, I would first assume stuck injectors especially on a car that has been sitting. The easiest and cheapest is to tap the injectors while someone cranks.
Kragen and others sell little gimicks that attach to the injector wires to tell you whether they get a signal. That might be a handy tool to decide whether a no-fuel situation is due to plumbing (pump, filter, clogged injector, stuck injector) or electrical (faulty DME)
Re-soldering the DME comes WAY down the list only when I know the DME is faulty. I would never do that unless I have verifiied with another DME this this is the issue. And even then there are more sophisiticated ways to test what's wrong with it.
Ingo