All my years of babysitting are finally coming in handy. My daughter's six month's old. She's been sleeping through the night since she was about two to two and a half month's old. Sorry, ya'll:)
First and most importantly--sleep when the baby sleeps or you won't. Try to do bath time and then a bottle. I find that the lavender bath relaxes and calms her, she eats and then it's "light's out".
Take turns sleeping if you can. Her first couple of month's were really hard. She's breastfeeding. So while daddy was sleeping (and I wanted to pop him) I was half asleep and trying to function.
That worked out pretty well. If you have a relative or friend who can come over for a couple of hours to watch the child so you can get a couple of hours that would help too.