Here are some good examples of various ways to get clearance from uncontrolled airports:
On the initial contact you need to let them know who your are, where you are, that you want to pick up your IFR clearance, and when you’ll be ready to go. Not much different than picking up a clearance at a towered airport.
Copying IFR Clearances
Just to get you started, here’s Garry Wing explaining how to copy a clearance.
Most of Steveo’s videos start with picking up a clearance. Here he is departing VFR, then picking up IFR clearance in the air. The pickup starts at 3:30 but he talks through departing an uncontrolled field so it’s worth listening to.
Picking up IFR clearance on the ground from KAUN through the phone
Picking up IFR clearance on the ground using the radio to phone patch through an RCO:
He explains the phone patch clearance at 14m40.
Picking Up Your IFR Clearance From An Uncontrolled Airport
This is an example of how not to pick up your clearance at an uncontrolled airport. The student IFR pilot gets the mechanics of the call right but makes a couple of mistakes. First, he should be familiar with the waypoints that are likely for his clearance. After leaving the airport and entering controlled airspace, you know that you are going to be cleared to a waypoint or VOR on the chart. So be prepared and know how they are spelled. Second, he accepts a clearance for takeoff in five minutes. The readback takes three minutes and there is no way he can get in the plane, do a runup, and take off in two minutes. The clearance void time means that if you are not airborne at that time, your clearance is void. He should never have taken off.
Hard IFR from Auburn KAUN to Oakland KOAK
Here’s how you make a phone call to pick up your clearance. He has a local number to call to get clearance and calls back when he is ready to go. His void time is also five minutes and he is off before that. The clearance starts at 100 seconds in, but you might want to watch the first part where he talks about the weather.
He flies out of this airport all the time, so he knows the local number. If you don’t know it you can call 1-800-WXBRIEF just like you do to get a briefing and ask for your clearance.