It's hard to write a resume and avoid discussing your situation, but it can hurt your chances. Gerrit Hall, co-founder of the great resume-analyzing webapp RezScore, has found that many applicants spend too much time focusing on their current circumstances rather than selling themselves:
Consistently, the biggest mistake we see is that people write a ‘me' focused resume. A primary example of this is the outdated objective statement — if you have the word ‘seeking' on your resume, you're writing a ‘me' resume. Employers don't hire you for your satisfaction; they hire you to fill their own critical need. Think of it this way. If you were in sales, would you ever say to a customer "Buy this item because I need the commission?" And if you were the customer, would you buy? A ‘me' centered resume says essentially the same thing.
Your job is to think of the potential employer as a customer. You've know they're a hot lead because they've taken the time to post the job — so someone is going to close the deal with them. How do you make sure they go with you? By selling to them like you would sell to anyone else. Figure out their pain points. Why are they hiring? Who have they hired in the past? What's their most critical need? And then go in there with your sales guns blazing; be the solution to their problem.
As we discussed this weekend, it's so important to put yourself in a company's shoes when you're applying for a job. Do your research on them and how they operate and remember to sell yourself rather than explain your situation.