How should I request an IFR approach at the end of a VFR flight?
Since you are already on flight following you are already in the system (ATC has a flight strip for you and a datablock on the radar).  Just call ATC and ask them for IFR to your destination.  Don't bother asking for a frequency change to FSS to file a flight plan, only to come right back on and ask to pick it up.

Austin Approach, Cessna 12345, request IFR to Austin

Since you have flight following ATC already knows your type, equipment suffix, location and altitude, so don't repeat it.  ATC should ask

Cessna 12345, verify you are qualified for and capable of IFR flight?1

and after you (hopefully) reply in the affirmative will issue:

Cessna 12345, cleared to the Austin Bergstrom Airport via fly heading 270, vector to ILS Runway 35-Right final approach course, maintain 2000.

If you are already in the approach environment you probably won't get a climb to an IFR altitude, you'll just be given whatever altitude they need for the arrival flow.
This method is faster than filing with FSS and easier for ATC (they don't have to approve a frequency change, wait around for you to file, then have you come back and wait further while your flight plan works its way through the system).  However, bear in mind that none of the &quot;extra&quot; search-and-rescue information you would give to FSS (fuel and souls on board, emergency equipment, filed alternate, aircraft color and markings) will be recorded.2
If you are landing at a towered airport you will not need to cancel IFR as the local controller will do this for you.  If you are landing at a non-towered airport you will need to radio ATC or FSS or telephone FSS to cancel your IFR service.

Footnotes
1This is a relatively new requirement after a non-IFR-rated pilot accepted an IFR clearance and subsequently crashed.
2Some facilities will have you rattle all that information off on the controller's frequency or a separate unused frequency and then mark that down in the flight plan remarks, just to have it on the tapes in case it's needed. If you're only 15 miles from your destination they probably won't do this but it's a possibility.