First, read the problem carefully. The facts and procedural posture of the case will become very important as you hone in on your research. For instance, was the case disposed of at summary judgment? did it get to a jury? are there evidentiary issues? whose burden was it to challenge the adverse ruling?
Begin the research. There are two ways to start. If the problem lists cases or shows some legal analysis in the fact pattern, then start researching by looking at the cited cases. If the problem does not cite legal analysis, then skip to step 3.
After you've completed step 2, the real research begins. The best way is to start broadly and progressively get smaller. Start with law review articles or treatises (AmJur and ALR are the big ones, and each state has different, more specific versions like CalJur or Witkin).
With a broad understanding of what the law is in various jurisdictions through the treatises and law review articles, you can now start looking at individual cases. Keep an eye out for cases that have similar fact patterns to your own, the analysis may be very persuasive to a judge when you write your brief.
Make sure you keep a log of your research so that you don't have to duplicate your steps. I keep mine in a legal pad, though notebooks work just as well. Loose leaf paper tends to get lost, so avoid distracting problems like that.
When you finalize your research, outline your thoughts, with particular emphasis on including case names and page citations to save time once you start writing. The outline should form the basis for your actual writing, and the headings for the outline should give a good idea of what your argument will be in each section.
Start writing. Everyone has their own style, so find yours. One way to write each section is using IRAC, which stands for Issue, Rule (statutory or judicial), Analysis, Conclusion. Make sure your arguments address both sides of the point, with far more emphasis on your own argument of course!
Ask someone else to peer edit your work. Whether it's a fellow student, TA, or instructor, a fresh set of eyes will always help in shoring up loose arguments and loose language.