I'm no expert (on my first android phone), but will try to help
So you want to get your contacts onto your new phone i presume.
Firstly, have you managed to solve the problem?
If not, then by 'not charging', do you mean that it's either bricked (incorrectly tweaked) or damaged (impact, water etc.)? Can you turn it on at all? (not just on battery but, for example, with the charger plugged in and switched on?)
From what I can tell by doing a few google searches, the contacts are store in a .db format, on the phone's internal storage, and cannot be accessed unless you're rooted
(and i'd presume that you need your phone to be working as well...) using the desktop based Android SDK ( In which format the contacts are stored in android? - Stack Overflow
So if I was you, I'd start with getting the phone working - even if you don't use google sync to copy them over, you could send them to your new phone via bluetooth (unless your new phone's an iphone, which, as far as I can tell, fails in this region because it's near impossible to send files to other phones - my gf's got one and it drives me crazy when we have to do simple things like send a file to each other and one of us doesn't have internet
) so you'd have to go with other options like emailing them or something... if that's possible...
Hope that helps