You can start the Trials of Thanos either via the Challenge Mode or by going to the free play computer in Story Mode and choosing them there. You can only get this achievement by playing the Trials in Story Mode.
When you play the Trials you have no partner and must play as Thanos through 20 waves of enemies. After you finish each wave you'll be rewarded with a white health pack to replenish a small portion of your health. If you die you will be brought back a few levels, but you will start with full health from that point, so it can actually be helpful.
The best advice for surviving the Trials is to use Thanos's jump attack as much as possible and if you are surrounded to use his heroic attack. The only other part that may be difficult are the enemies that use shields and are therefore harder to hit. On those waves block (hold down the button) in the middle of the platform and when there is a break in their attacks use your charge attack (hold down ).