Each of the five drones has five levels. Each starts at Level 1. To get to Level 2, you need to spend 2500 points; Level 3 needs 3000; Level 4 needs 3500; Level 5 needs 4000.
That means to get to Level 5, you need to spend 13,000 points on a drone. Every time you finish a level (pass or fail, doesn't matter), you're awarded a number of points that equals your multiplier. Finish with a multiplier of x689? You get 689 points.
After fully upgrading the first drone, you'll get the Specialist achievement.
Upgrade all five drones (which you'll unlock by beating bosses in Adventure Mode) to get this achievement. You'll need 65,000 points to fully upgrade all five.
If you don't want to rush this achievement, you'll accrue enough points playing Adventure Mode to unlock it naturally. If you want to grind it out, the easiest way to get points is through the 'Evolved' levels. Level 35 ("Endurance") will net you the most points in a single shot, but it's a 10 minute level.