I have been offered a PhD position with full funding from my backup university, however they gave me a deadline of April 5th to either accept or decline their offer. I am still waiting for replies from other more prestigious universities. What should I do?
There are generally two things you can do - and you can and should do both at once.
1) Try to extend the deadline. Contact the people who've offered you the position and ask for extra time. They aren't going to give you an infinite amount of time, but ask for ideally how long you'd like - say, an extra two weeks? - with the expectation that they might be willing to give you less time than that. They might say, "Sorry, we can't extend the deadline" or "we can only extend the deadline X days" - but hey, that little bit of extra time could be very important! If you are polite it can't really endanger your offer at any place you'd want to dare attend, and they could always say yes. Just be 'reasonable' in your request - asking for an extension until days before the academic year starts won't be granted and might encourage them to give you no extension at all.
2) Contact the programs you haven't heard from and let them know you've received other offers, and that you'd really like to still consider their program - but you are on a deadline. Inquire if you'll be able to get an answer from them before your deadline. Everyone in the process understand that candidates who get in anywhere often get multiple offers, and deadlines don't always sync up. If you've had contact with a prospective adviser at the institution, consider CC'ing them as well (on the chance that they have already made a decision but the institution is being slow).
There's not really anything to lose if you act professionally, and potentially a great deal to gain. You could get an extension, you could get an early response from other programs, you could find out responses were sent but did not reach you - or worst case scenario you end up in exactly the same position as you are in right now.