Luck and underhanded tactics are your friend in the Battle Royal. I feel like the following are important tools:
1) Strong priority moves. There's moments where you'll find yourself in a 3v1, but that's not always how it works out. Sometimes, the NPCs gang up on each other. For that, you can use strong priority moves like Scizor's Bullet Punch or a boosted Shadow Sneak from Aegislash to steal kills from your opponents when they get another NPC's Pokemon low on health.
2) Protect. There's a reason why Protect sees so much use in Doubles; it's invaluable for avoiding attacks while another Pokemon takes out potential threats, and the same logic works here. Half the time, if I've tossed my Feraligatr into Master and it's up against Verizion, there's going to be something like Heatran or Azelf waiting on the sidelines and itching to kick its ass. Sometimes you have to let the NPCs sort themselves out and that's when you let them have it.
3) Bulky attackers. Aegislash and Foul Play Umbreon are some of my favorites to run in the Battle Royal because they're excellent switch-ins when my lead is being threatened. They can take some serious hits, give them back, and with Shadow Sneak, Aegislash can even steal some kills while it's at it. They can save your life.
Additionally, I would avoid overuse of stalling tactics and status moves -- especially on Pokemon that don't have a lot of bulk. If you have fast, frail leads that like to set up or bulky Pokemon that enjoy tanking attacks while employing tools like Leech Seed or Toxic, they're not going to have a good time in the Battle Royal. In the Battle Royal, it helps to have Pokemon that are either fast and strong without much need for set-up or strong and bulky so they can take an attack and fire back.
