The 10,000 Solar required for this achievement will come relatively quickly once you've got a handle on how the game is played and have found an Awesomenaut you consistently do well with. In a shorter match, I averaged between 1,200-1,600 Solar in roughly 6 to 8 minutes. If you like, you can prolong matches by not attacking turrets and focus on killing other Awesomenauts to greatly increase your per match earnings.
After much experimentation between active play and idling, I've come to the conclusion that, unfortunately, only actions you yourself complete will contribute to the three Solar collection achievements. Below is a breakdown of the values for each way Solar is awarded in a Bot Match. I'm not sure if online matches differ in any way, and bot difficulty doesn't seem to matter.
+1 Solar for each silver coin collected
+1 Solar every two to three seconds you're alive (earnings paused during respawn)
+3 Solar for killing a Creep
+5 Solar for killing a Droid
+10 Solar for killing a Super Droid
+20 Solar for killing a Solar Boss
+40 Solar when you end an enemy Awesomenaut's killing spree
+70 Solar when you score a kill on an enemy Awesomenaut
+100 Solar when you destroy a turret
-25 Solar when you are killed
It is unclear if the passive Solar and collected silver coins contribute to this achievement, but those amounts are fairly negligible anyway, so I'd suggest focusing on the performance based earnings.