In order set a new shot record you need to do 1 of 3 things:
1) Beat a record you have previously set on any hole in each of the 4 locations.
2) Beat a record a friend(on your friends list) has set on any hole in each of the 4 locations.
3) Beat the world record set on any hole in each of the 4 locations.
The record you beat can be a number of things including, but not limited to: Drive distance off tee, closest approach to the pin, or longest putt sunk, just to name a few.
This achievement is nearly unmissable as you will unlock better gear and are likely to replay many of the courses to achieve the elite ranking. If you want to speed this process up, you can do the following:
1) Select a short course in a location (Just par 3's/4's/etc)
2) Intentionally do a very short drive off the tee (make sure to land on the fairway, otherwise it will not count)
3) Finish up that hole, and then the remaining holes in the course.
4) Load up that same course again and drive the ball longer off the tee (again, making sure to land on the fairway).
5) If done correctly you'll receive notification of a new record set. You must complete the course for the record to save. Do this again on the other 3 locations and the achievement is yours.