This is how I kill Drume solo and it always works for me:
1. There are 3 Usurper Commanders that you have to kill before Drume spawns. The Commanders spawn with other monsters (the warlocks are the biggest damage dealers, try to avoid them). The best strategy is slowly luring and killing the commanders 1 by 1. If you get overwhelmed, run south west, there are a lot of trees there that you can use for kiting.
2. Don't kill the last Usurper Commander in a random spot. It is best to finish it off in the south. Make sure you keep a Lion Commander close during this fight, as it will stay there once you kill the Usurper. This is important because sometimes (especially if you take a bit too long) Drume will manage to kill all the Lion Commanders and the battle will be lost. Keeping 1 commander away from Drume won't let that happen.
3. Drume will spawn close to Kesar, always in the same spot. Sometimes you will be lucky and he will target Kesar straight away, sometimes he will attack the other monsters. You don't have to worry about it if you have a Lion Commander safe in the south, away from Drume.
4. Always stay as far as possible from Drume and use an energy bomb on him. This will keep him in the same spot for most of the time. If he retargets, run away until you don't see Drume on your screen (run a bit further just to be sure).
This strategy always works for me and makes killing Drume quite easy. Good luck!