While the other two solutions tell you the bare minimum regarding how to get this achievement, I thought I'd give a few useful pointers.
First, you can do this with a dummy account and all on the same xbox. The only requirement is that you be XBL friends with the account in question.
Second, you can find all of the records through the in-game leaderboards. Every course and every hole as its own set of records (I'll get back to this a little later). Scrolling through all of the records is pretty simple too.
Third, you can't be the first person to set the record on your friends list. You have to knock somebody (who has already set the record) out of the top spot, have someone beat your new record and THEN take back your record [I know this due to trial and error].
What I did to earn the achievement is scroll through all of the records until I found a record that hadn't been set by anyone on my friends list and :
1) Used my son's account to set the record;
2) Beat his record;
3) Had him beat my record; and
4) Beat his record again.
A really good spot to do this on is Hole 1 on the Coyote Canyon Course, and on Amateur difficulty (i.e., easy pins). I have many friends who had played this game (most aren't playing it anymore) and so it would be hard to get this achievement through natural play.
Hole 1 is a Par 3 and so everyone just naturally goes for the green on the first shot. However, I noticed that there was a record for best shot within 150 yards - the only way to meet this criteria though is to have an absolutely horrible shot off the tee [there is another record for best shot within 75 yards which supersedes this one apparently (if you are between 76 and 150 yards you are only counted in the latter leaderboard)].
Anyway - no one on my friends list had set a record for the "best shot within 150 yards" record leaderboard, and so I only had to worry about my son's account and my own. I proceeded to hit a horrible tee shot (using a SW), and then hit the ball on the green but as far away as possible from the hole. FYI - to beat the record using this method you have to land on the green but closer than your friend's record. So, you don't want to set a really good record that you will have a hard time beating.
My only other tip is that if you use this method, use a 7 or 8 iron on the second shot and place your shot marker so that you are aiming for the far side (i.e., passed) of the green. You will be shooting from the rough, so if you are aiming to hit the ball to have it land on the green, you will likely come up short and land in the bunker (just before the green).
Using this method is not only relatively easy, but you will save a lot of time as you will be starting your attempts on the first hole of the course.