If you already have your DR residency, you can apply for citizenship. If you don't, you have to start residency back in home country to get it and then, after 6 months I believe, you can apply for citizenship.
Exactly how I did mine. First you have to get the temporary residency to get the process started. Back when I did it I didn't have to go back to the States to start that part but it seems that now you must start it within your home country because some of the paperwork can no longer be completed here or mailed. I think the big thing is because you must stop by a Dominican consulate in your home country. So I was lucky that I did not have to do that.
After you have residency for six months you can begin the citizenship process. Until your citizenship is ready you must keep your residency up to date. That means renewing it every year and if it takes too long to complete your citizenship you may get permanent residency after four years. The first one must also be renewed in one year and then after that you renew it every four years. My citizenship came rather quickly, but my understanding is that it takes much longer now, mostly due to Interpol delays.
The process is relatively the same for the permanent residency and the citizenship as it is for the initial temporary residency. You need medical exam, drug screen, etc. but they are basic things. Residency renewals don't require any of that. The difference with citizenship is they do the Interpol check and you have to take a test which you can prepare for and is pretty easy and the oath which is also easy.
I did the entire process on my own with help from my wife. We didn't try to bribe to push anything along faster than it should be but I let her be the persistent one and do the talking when it seemed like we were getting the run around. When I did mine the migration department was a circular mess of disorganization. Not sure if it has gotten better or worse with their recent changes.