Basically, what you want to do follows these 4 rules:
- Avoid "boring" kills
- Preserve their hearts when possible
- Rack up bonuses
- Get rid of the lights
Boring (+10) kills
I've detailed all the different ways of killing as recognized by the game here. Basically, avoid the +10 kills whenever possible. To recap, these are:
- Non-headshot kills with a weapon
- Neglecting or throwing a held (alive) human
- Killing someone with a blunt thrown object
So if you have to use your weapons (you will have to), aim for the head. If you grab someone, slash him in half or do an execution. The last one isn't really a problem; an enemy will have to be significantly weakened to die from that.
Preserve their hearts
This doesn't really apply to the beginning of the game, where an Execution is generally the best move (just behind a +20 kill with a Darkling assist, and then eating their heart), but later on, enemy hearts will be worth more (up to +30 essence).
Hearts are destroyed when executing an enemy, gibbing them with Gun Channeling, or when they're sucked into a Black Hole. It's possible for enemies to be killed by Black Holes without being gibbed.
When enemy hearts start being worth +10, a non-boring kill (+20 and up), combined with eating the heart, will be worth just as much as an execution (+30).
You should follow this tip especially if you have the Limited Edition DLC, which adds a bonus 20% Essence for each devoured heart — a maximum bonus of +6 for hearts that are worth 30 Essence.
There are a couple bonuses, and obviously you should strive to get them whenever possible. I'm going to list all that I've gotten so far, which should hopefully be everything.
Double kills with thrown objects (+20) - There's definitely a bonus for double kills. Not sure about triple, never done that.
Darkling Assist (+10) - Finish off an enemy that's being distracted (or killed) by your Darkling. If you knock the enemy around without killing him, the Darkling will detach and you won't get the bonus.
Fast Switch (+10) - I'm not entirely sure about this one, but it has something to do with rapidly using different weapons.
Fast Killer (+10) - Awarded for rapidly racking up kills. I usually get this when I'm being backed up by the Swarm or a Black Hole, or when using Gun Channeling.
Gun Channeler (+20) - Kill a certain number of enemies while Gun Channeling is active. Don't know the specific number.
In the Cloud (+20) - Not entirely sure about this one, but I'm pretty sure you get it for finishing off all enemies that are being eaten by your summoned Swarm. Summoning it on one enemy and then finishing him off should do the trick, but that's obviously a waste.
Irresistible (+20) - Your Black Hole does a certain number of kills; I rarely ever got this. Enemies can resist a Black Hole, so make sure you shoot or slash nearby enemies to stagger them.
Blinded by the Light (+20) - Has something to do with getting kills while standing around in the light. I only got this in Vendetta mode, where you aren't quite as helpless in that situation.
This isn't nearly as important as the rest of these tips, but in the singleplayer campaign, shooting out lights (or otherwise disabling them) is worth a tiny +1 Essence bonus. Might as well get them, with the added benefit of not being blinded.