First of all things aren't that simple.
HE needs to take care of everything on his end. Since he was not there when the child was born, the Army may require a DNA test to prove paternity. Did you put him on the birth certificate as the father? If so things may be 1% easier for you. If not you WILL need the paternity test to get any coverage.
He is kind of blowing off his responsibilities if he hasn't taken care of all of this already. For him to do anything, you will need to mail him a copy of the certified birth certificate that has his name on it as father. HE will need to take that to his admin officer and add your child to his Page 2 as a dependant. After that is born he will need to complete a DEERS enrollment form that will put her into the Tricare system. He will also need a copy of the childs social security card and all other information like your full name, the childs address and your address. He will also need to get an ID Card application done because in order to receive medical care the child will need to be taken on base. The child will need an ID Card done because he in deployed and will not be able to take her himself. Normally children do not get ID cards until they are 10. You will need to jump through hoops too to get onto base because you are not and will not be the childs sponsor that is your boyfriend. The child also CANNOT sponsor you because the child is a baby and can not be help accountable for your actions.
Now if you didn't put your boyfriend down as the father on the birth certificate so you could receive aid like welfare and other things, you are probably not going to be getting any Tricare coverage until he gets home from deployment. He, you and the baby will all need to do a paternity test first to establish that he is really the father. Once it is determined that he is in fact the father than he will need to go do all the paperwork I mentioned above.
If you want your life to get easier, you need to get married. Until then your care for your child is going to suck because you will get hassled everytime you try to get onto base because you will never be allowed to have base stickers or a military ID for yourself. That child is going to need to go to the doctor nearly on a weekly basis for several months when it is young. You and the father need to figure out what you are going to do quickly ot you are going to need to get the baby on public assistance for its medical care.
Sorry the story isn't nicer. I have seen this situation a million times and it never gets easier. There are a million women out there that claim a military guy is the daddy and in reality he isn't. That is why the system is set up so the military does not get stuck paying the bills for everyone.
I hope things work out for you.