The Vanquish Streak Boast can be obtained by having the highest Vanquish Streak in any gamemode. The best method is to use the Cactus in Gardens and Graveyards. As the Cactus, always hide in the back as a sniper picking off enemies in the distance. This will get you plenty of Vanquishes while preventing the occassional death. Also, every now and then, use your Drone to get a couple more Vanquishes aswell.