Many foreigners have been working here for many years without a "residencia" which permits you to extend your stay beyond 90 days and to work here. Still it is illegal to do so. You can hire a lawyer to help you get your residencia at a cost of about $1,000 US, or you can do it on your own by visiting the Immigration offices in Santo Domingo.
If you do it on your own, the cost is a few hundred dollars for medical exam, police report, translation of birth certificate, etc. My last friend who went that route calculated that it took hom 15 trips to Santo Domingo, and cost almost as much money as using a lawyer. It took him one and a half years from start to finish.
You will then have to repeat the process within 2 months, or one year, to get your "definitiva" or permanent residency. The important thing is that once you file your papers you will get a "comprobante" which will permit you to stay while you are waiting for the process to be completed.
Unless you have unlimited time and patience, I recommend that you use a lawyer. Not just any lawyer, but one specializing to some degree in that area who can supply references of foreigners you can contact that got their residencia through him.
Norman