Once you arrive at the museum, walk all the way across to the loading area where you'll hook up with Niko at the goods entrance (Niko is friendly to Johnny and is clearly unaware of his involvement in Roman's kidnapping). The meetup cutscene between Johnny and Niko is the exact same from the main story, as is the following cutscene when the group is fired upon from above during the deal by Luis Fernando Lopez. Just like the main story cutscene, some of the other party's members will be killed.
In the hail of gunfire, Johnny grabs the cash and makes a break for it. From here, things get to be a bit different from Johnny and Niko's perspective. You'll start in the upper left hallway, while Niko is on the right. You can actually hear Niko using his taunts during the fire fight as he does when you're playing as him. If you quickly go to the railing and look down at the museum floor, you can see Niko fighting his way to the exit. (Niko can be killed by your weapons, which will fail the mission and trigger a phone call with Ray about Niko's death.)
Just as you did with Niko, you make your way down the hallway killing the foes in your path. However, once you get to the staircase leading down, you'll realize that a ton of the enemies are already dead (all thanks to Niko), making this fight a lot easier for Johnny than it was in the main story (especially if Johnny uses the assault shotgun, which is one of the most powerful guns in the game; the sniper rifle is also very good for taking down targets on the other side of the museum). Johnny observes all the dead bodies and comments that he'd hate to have this Niko guy after him because he's "deranged" given the evidence all over the building. Instead of going the way Niko did in the main story mission, you make a sharp left after descending the staircase and head down through where you originally came into the building at.
Outside, Ray's goons are lying in wait, taking cover behind their black luxury sedans. After taking them all out (you have the option of calling in your backup, though it's a good idea to call them in before you exit the museum), two or three cars will make off. Chase down and kill these foes (if you can't locate your bike, there is usually another vehicle or two parked near the museum entrance), then evade the police, as you've likely gained a 2 or 3-star wanted rating by this point. (Note: although Johnny says he needs to kill the drivers before they report to Ray, there is no time limit for tracking them down as they simply keep driving around the general vicinity of the museum.)
Once the wanted level is removed, Johnny will call Jim and ask to meet up so Johnny can give him something (the cash stolen from Ray). During the trip over, Ray calls and says he's been informed by Niko that the money was snatched in the midst of the chaos. Johnny pretends he just wanted to escape with his life and had no idea what happened to the money. Ray then threatens Johnny, stating that if he's trying to pull one over on him, he'll pull Johnny's teeth out of his head. You make it to Jim and give him the diamond money stolen from Ray, saying "fuck him" and justifying keeping it. Jim's on-board and agrees to take it from here. After that, the mission's successfully completed and Johnny is $10,000 richer.
Shortly after completing the mission, Johnny will receive a text from Jim which unlocks the next mission as the endgame approaches.