Fight Baron Blade at the Wagner Mars Base and have him incapacitate Legacy. Do this with a team of five with Legacy and several heroes that use items or ongoing effects (for example, Tachyon, Tempest, Unity, and Absolute Zero). Have Legacy use Lead From The Front. Note: Card-drawing heroes such as Tachyon and Visionary can be used to ensure you get to Lead From The Front in time. Tachyon and Wraith can also help with the Wagner Mars Base's self-destruct sequence. When Baron Blade draws Devious Disruption, the damage he does counts all ongoing and equipment cards. Also the damage counts as coming from Baron Blade, and builds into several huge bursts. Redirect that damage to Legacy until he dies.
Easy "Extreme Hostage Situation" achievement
Use Baron Blade at Insula Primaris. Have Legacy in the first slot, Fanatic, Haka, Ra, and then Tempest in the last slot. This will allow you to do at least 12 damage per round and 17 damage per round once Obsidian Field comes into play. You will need lots of damage to take out the Baron's minions and platforms. Select Tempest's attack order carefully to get a few more points of damage. When Volcano appears, have your teammates all turn on Tempest and knock him out. Next, use Tempest's knockout power to give your teammates immunity to fire damage. Having Volcano damage all enemies and no damage to you makes this easy.
Easy "HAKA CRUSH" achievement
Have Fanatic's hit points drop to 10 or less, then play Wrathful Retribution. Because she has only has a single copy of Wrathful Retribution in her deck and little card-drawing of her own, pair her with someone that has card-drawing. Use Bunker and his Ammo Drop. He will not require a card drawer. Load his Omni-Cannon with ten cards then proceed. Use Absolute Zero, Ra, Legacy, and Visionary. Bring Absolute Zero's equipment trifecta into play. After he has his equipment ready, he will apply damage boosts coming in and going out. Combine this with Ra's fire damage boost, Legacy's damage boost, and any other damage boosts available. Have Visionary Twist the Ether before a large damage source (for example, Omnitron's 15 point bombs) and he can reach the 20 point range. Note: Consider Tempest as a fifth for the healing and to have Into The Stratosphere for any ongoing undesired effects.
Easy "Mobbed by Paparazzi" achievement
Incapacitated abilities do not qualify as "powers". Have Absolute Zero, Ra, and Tachyon against Ambuscade in Insula Primaris. Absolute Zero's equipment allows him to heal from fire damage, Ra's ongoing gives him immunity to fire damage. Tachyon's Synaptic Interruption allows you to choose which villain or hero to redirect damage to. As soon as the volcano appears, it will be easy. Also consider using Wraith for her Smoke Bombs. With the DLC, fight The Dreamer with Legacy, Omnitron X, and Ra in Megalopolis. Omnitron X is very useful with his plating and ability to heal teammates and damage foes with effects that are not powers. Legacy and Ra boost Omnitron's damage. You can also take Wraith instead of Ra for damage reduction and redirection; or Tempest to have another hero with ongoing damage resistance and incapacitated power.
Easy "The Super Scientific Tachyon" achievement
Tachyon has to discard five Burst cards in a single turn. This can be done by Tachyon herself with her card-drawing and action-granting actions in her core deck. However extra card draws with heroes such as Legacy or Visionary make this faster.