It's worth removing the battery then power up on mains electricity alone and see if it works then.
Go into your Contacts and tap for the menu which allows you to Export. If you have a micro SdCard in the phone, export to that. If not, connect to a PC or laptop or buy a SdCard.
When they're exported they will be in V card mode and you can view them on a PC. When you get a new phone, use that card or USB conenction to import them.