I decided to take a crack at this myself. What I found most frustrating about all the video guides is that everyone is whipping their camera around like they're in the middle of a multiplayer match, and it looks like they all got the achievement by accident.
What I think makes this the easiest to explain is to use references for direction based from the mainframe you're standing on. The thing bangs around, rotates, and swings, which makes it super hard to use anything else in the environment for reference. We'll call Woods' side south.
There are 16 total enemies that can be exploded by 7 total barrels.
0:00 When playing the mission through normally, after you attach the winch you are made to look west at the first group of enemies. Your exact view may change based on checkpoints.
0:05 The first explosive barrel is southeast on the second story. Three enemies will cluster by this one.
0:12 On the floor directly above the last, directly east out of the mainframe, is the next. Three more enemies will come out near this one.
0:19 - 0:26 West from the mainframe and up two floors is the third barrel. I believe only two enemies stand near this one.
What follows is several seconds of the mainframe swinging around wildly. 0:31 Look directly out the east side, and right after 0:33 the mainframe smashes into the wall, two enemies will stand near it.
0:37 This one is hard to spot but will net you two more explosive kills. It is north out of the mainframe by the time you can shoot at it, and is directly right from the previous barrel on the same story of the building in the corner.
0:42 Out the west side of the mainframe are two more barrels on the same story of the building with two enemies each.