I was going to make a guide for this when I first got it, but I had so much trouble that I wasn't really sure the best way to put this.
Depending on your views, this could be easier or harder than other achievements. On the one hand, compared to most Ultimates, you're still able to move around and attack during it. Most Ultimates are a one-off attack in one location. This means that while you don't have to depend on 4 people being right on top of each other, it requires a little more input on your part. Also, Genji's callout seems to be quite a bit more hindering than other Ultimates. It's louder and it goes on for about 2 seconds before his ultimate even starts.
Genji's Ultimate causes him to pull out his sword and deal a fair amount of melee damage in a wide arc. The swings do 120 damage. You can also still use your other abilities while using Dragonblade. One issue that set me back a lot was that I always forgot to use Deflect. Don't be afraid to use it since people will try to focus you down. Also, your Swift Strike attack resets whenever you get a kill, so you'll want to get used to throwing that into the rotation.
I personally have had a much easier time going or most of my achievements in Escort, but this could vary for you. In Escort, you have the benefit that most enemies are going to be facing the same direction. The downside is that there are a lot less flanking locations. This is also true for Assault maps (where one team defends and one attacks).
On the flip side, Control maps are more likely to have a cluster of enemies, but they're probably a bit more likely to be checking every direction.
So far, I have had the best luck getting my multi-kills on King's Row (with 3/3 of them so far). I believe this is due to the very tight corridors, and the drastic differences between high and low points compared to other maps. This is especially true during the first checkpoint of the escort. The Payload has to weave down very tight streets.
Enemy team composition also plays a part. Tanks are going to waste a lot of your time if you focus them down. Only go for them if you're hitting other people. You're going to want to start the Dragonblade by taking down their healer if they're alive, then Swift Strike people to get behind them. If you can get a Zarya to group them up or a Mei to freeze them, that's icing on the cake.
I'll cut down this video later, but here's my unlock (after 13 hours of going for it). This was a pretty sloppy Dragonblade as I tried to kill the teleporter which was no longer there.