I'm not sure exactly what you are starting with, but basically you will need to attach boards (2x4's for example) to the ceiling joists to use as nailers for the drywall. If this false ceiling needs to be considerably lower than where the current ceiling is, you can nail together a structure that hangs down from the ceiling joists and then attach your nailers to it. Do this by opening up the existing ceiling enough that you can attach 2x4's along the length of each joist at 16" intervals. Extend these up so that you can nail them each with about six nails, three at the top onto a truss cross member and three at the joist. The length to extend them down below the joist is to where ever you want the new ceiling to be. Then attach 2x4's (the longer the better) to these pieces, flush to the bottom end, going the length or width of the room. You will want to attach spacers to the tops of these too, starting at one side of the room and measuring 16" on center from one nailer to the next. Do a lot of measuring and make sure this distance is accurate and consistent around the room. Use an extra nailer to make up for a spacing that doesn't come right out to 16 inches so that you have no span greater that 16".
Once all your boards are up and accurate, attach your 5/8" fire barrier drywall to the nailers using screws that are 1-1/2" to 2" long and spaced no more than 12" apart along the length of the sheet. Mud, sand, prime and paint and boom!
Amlove32