Since getting to the Evacuation Ship is fairly straightforward, we're going to focus on destroying it. If you are on the losing team, keep in mind you can earn the Refugeachievement by surviving and getting out alive.
If you're on the winning team, there are two ways to emerge victorious in the Epilogue. First, you can kill all the opposing team's Pilots, earning the I killed Them All achievement if you get all six. One of the most effective ways to do this involves heading to the Evacuation Ship and cutting them off.
The second way to come out on top in the Epilogue is to destroy the Evacuation Ship itself. This is much easier than tracking down and killing all six Pilots individually. To pull this off, you generally need three Titans engaging the ship. You can pull it off with two, making up for the loss in firepower with help from Pilots or by using the Atlas Titan's Damage Core.
For more strategy related to Titanfall, be sure to check out the Limited Edition Strategy Guide, authored by professional gamers.