This is the most difficult achievement in this game because it takes too much time to achieve. The developers are developing the game on a new engine, so this Unity version of the game will not turn into be a full game and will stay as a DEMO, so you got only 1 stage to grind an achievement that was meant to be unlocked while playing the full game. The shortage of stages makes the task pretty obnoxious and repetitive.

1 - You don't need to finish a stage to get your kills counted.
2 - Higher difficulty doesn't seems to raise the number of enemies

First, play the game normally and get fetch some skill points. Upgrade your Physical Skill Tree until you reach the rune "Phanton Strikes". Then, upgrade Fire Skill Tree until you reach the rune "Rejuvenating Flame" and the rune that gives "+65% increased chance to draw a Fire Ability".
If you got spare points, you can spend on other runes (mostly on the Defensive Tree), but do NOT get runes that "+65% increased chance to draw..." of other trees besides the Fire Tree.

you are going to select the easy difficulty (since difficulty doesn't increase the enemy rate), and every time the skill menu appears, you need to search for the "Fafnir's breath" and get it 6 times, until you get the purple chest to upgrade it, then fill your skill tab with lvl1 fire skills. Move to the center of the map and relax. With that strategy, you are going to kill any automatically (even the boss) without having to move, and can do other chores while grinding, you will only be required to skip the level-up menus when they appear.
You should be getting around 15,000 kills per try.

The "Fafnir's breath" when upgraded reaches the entire map so you don't need to chase the enemies.
The "Rejuvenating Flame" heals you every time 5 skill are cast based on your max health, that is why you fill your skill tab with fire skills. You can also benefit from increasing your max heath and the cooldown of abilities.

This skill when upgraded also reaches the entire map, but you wouldn't benefit from the "Rejuvenating Flame" rune healing if using Ice skills.

Funny question. They both work together when upgraded, but upgrading 2 skills is less consistent and unnecessary.
If using "Phanton Strikes" they can trigger so fast the enemies die on the spot without getting close to you, so you end up not leveling up, which is excellent, because you won't be required to skip the upgrade menus anymore and just go AFK. But there is a problem, when "Phanton Strikes" triggers, the game can freeze with the amount of casts.

So, 20min per section is a lot of time, and the game as a DEMO is pretty repetitive. When you level up and unlock some runes, the game on the lower difficulty becomes quite easy, so no matter where you walk, you kill the enemies effortlessly.
What if you could speed-up the time in-game?
With "modeengine" you can increase the speed of the game. It will not make the game easier, but will make the faster and help your grinding.
https://modengine.net/