UPDATE:
In addition to the below, I also note that you also earn kills while offline/idle. I was only a few thousand kills into the achievement earlier, but when I logged in after being about 10 hours offline, I jumped to 10,200 monsters killed. The achievement unlocked rather quickly once in the game proper.
The rate of kills may be dependent on your DPS versus your highest area/lvl reached.
- - - - - - - - - -
An enemy is killed when its health is depleted. You are rewarded gold, with which you can use to upgrade your heroes, to increase your damage, to kill more enemies, thus continuing the vicious cycle.
Bosses count as a regular enemy, in addition to having its own achievement:
As you defeat monsters, you will progress through the following in-game achievements, symbolized by a regular white skull with no jaw close to the bottom of the achievement list:
* Killer (Kill 100 enemies )
* Butcher (Kill 1,000 enemies)
* Executioner (Kill 10,000 enemies)
The achievement will unlock once Executioner is unlocked.
Tracking of progress in-game can be done two ways:
1) Selecting Executioner (it'll show progress as # of kills and % to completion)
2) Going to the in-game stats page, which has a "Monster Kills" counter (sixth one down)
Enemies in lower-leveled areas have less HP, and are quicker to kill. In addition, upgrading your heroes to increase your DPS will also make enemies die quicker. However, enemies have a respawn time of half a second. No matter how fast you kill a monster, you have a minimum of 0.5 seconds between kills. In addition, low-level areas give less gold than higher areas. A recommendation is to advance normally, building up your DPS. Once at a higher level, you can drop back and work on bosses for Boss Exterminator. Once that is unlocked, you can either harvest the boss for coins if you can kill them fast enough. Then you can either move onward, getting kills as normal, or grind out an easier area that may take a fraction longer than half a second. At that point, your DPS isn't being wasted while still ensuring swift kills.