What must I do for “Destiny Dominated?”

TL;DR Enemies are more difficult, no adjustment to pop rates.

According to Answers from the Dev Team

Reckoning offers three difficulty levels out of the box for players of all experience and skill levels. We’re hoping to cover a broad enough range with these difficulty levels to satisfy everyone from the casual gamer to a hardcore action gamer. When crafting the gameplay and tuning the moment-to-moment experience of Reckoning, we thought a lot about how to appeal to both of these sets, which is why we ended up with a combat system that allows fantastic moves without memorizing combos, yet still lets players find interesting attacks to string together.

Obviously, hit point amounts are an easy way to start scaling difficulty, but in Reckoning we try to go beyond that since there's more to being challenged than just being able to take fewer hits before death. A skilled player that puts the difficulty level up a notch has already mastered dodging and blocking, so what's the point if they still don't take any damage? Likewise, a more casual player may be able to handle the basic mechanics of combat, but might instead have trouble dealing with the pace of combat itself.

So while we certainly take advantage of adjusting hit points with difficulty, NPC's in Reckoning scale the frequency of their powerful attacks as well as change how many can be attacking the player at any one time. This isn't to say that easier difficulties won’t see powerful attacks, but a player choosing harder difficultly levels may also have to contend with several of these attacks from many directions. – By Michael "Dawesome" Dawe, Senior AI Programmer

Further, according to this

Hey guys, Justin Perez here, one of the Systems Designers here at BHG. When you go up to the Hard difficulty level, we do increase the damage and health of enemies across the board (and decrease when you go down to Easy). In addition we're also adjusting the number of attacks a player can be subjected to at once. In the simplest terms what goes on behind the scenes is this: Every NPC attack in the game has a weighted number value, and we allow all active attack weights to add up to X. On Hard, we let that X be higher.

To give you a real in-game example: on Normal difficulty if you fought a troll and a couple boggarts, when the troll does his big charging overhead slam, it basically locks out other melee attacks from happening at the same time. The player is left just to focus on dodging that troll's charge. However on Hard those boggarts are free to launch into their basic combo attack at the same time the troll is charging. So it's more stuff that is going on at once the player has to be aware of.

There are no adjustments to loot drops or experience based on difficulty.

(emphasis mine)

The "general" requirements for these 5 achievements are (According to Giant Bomb's achievement list):

• Message in a Bottle - Located all eight message bottles in Gallows End.
• Keep on Rising - Gravehal Keep has been fully restored.
• Give Her a Hand - Found Aubrey Gilcrest's severed hand.
• Manic Pixie Dream Elf - Wooed Rast Brattigan.
• Exterminator - Killed 50 Scavs.

I'm assuming these are related to specific quests in the game, but I haven't encountered these quests yet, so I don't have more detail than this.