Upon starting the game, every player is automatically assigned the Objective: Conquer Hoxxes IV assignment. Successfully completing the ten Missions indicated by the Assignment (in order) will finish the assignment and unlock this achievement.
The mission you need to perform is indicated by a white diamond shape in the menu, which you can see both on the planetary map (soloing or hosting the mission) or in the current games list (if joining a mission). You may see multiple white diamonds on the map/list -- any of these mission will do, so select the one with the difficulty you're most comfortable with (more on this below).
Starting out in this game, you can only access Mining Expedition Mission types, so you can't miss assignment level one. The rest of the mission types are unlocked as you complete the other assignments. They are, in the order needed:
1.) Mining Expedition (Unlocks Egg Hunt Missions)
2.) Egg Hunt (Unlocks On-Site Refining Missions)
3.) On-Site Refining
4.) Mining Expedition (Unlocks Salvage Operation Missions)
5.) Salvage Operation
6.) Egg Hunt (Unlocks Point Extraction Missions)
7.) Point Extraction
8.) On-Site Refining (Unlocks Escort Duty Missions)
9.) Escort Duty (Unlocks Elimination Missions)
10.) Elimination
Upon successful completion of each step, you also be awarded credits, crafting materials, and new mission lengths and complexities.
Note: It doesn't matter what level of Hazard, Complexity, or Length the Mission is when going to complete assignments, only the Mission type. Therein, select the easiest levels you can when starting missions for easy completions.