For this Achievement, you need to complete all 6 Missions of the second Lost Episode DLC. You will unlock Mission Mode once you complete the story part. These are the Mission tasks:
Mission 1: Perform 5 special attacks.
Pretty easy, hold B to knock him down, then press Y to use a Special Attack. Use 5 Special Attacks to complete this mission. Note you have several Matches to do this (2 or 3, not sure) so you have more then enough time to do this.
Mission 2: K.O. your opponent with his Super Combo Gauge maxed out.
Your opponent will have his Super Combo Gauge full at all times, so make sure you don't get too close to him. Or he will use a Special Attack on you. Basically all you have to do is spam B at the beginning of the fight, use the Y attack and attack him with A and X for the rest of the Match. This should so the trick.
Mission 3: Get a K.O. in 40 seconds or less.
This is one of the hardest ones. As stated you only have 40 seconds to K.O him, which is pretty hard. Use RB at the beginning to get close to him, spam B until you knock him to the ground, press RB twice and use a Special. Once he get's up try to hit him with yet another B spam, and then leave him be. Back off a little bit, and use the A and X method. Prepare to ragequit, this will be frustrating.
Mission 4: Get a perfect K.O.
You spawn without any form of health, so any attacks that hit's you kills you. So when the fight begins, press RB so you roll towards him, once you are out of the roll spam B. Keep spamming B until he falls/flies to the ground. Once that happens, press RB twice, so you're right next to him and execute a Y attack. Doing so will take 1/3 of his health. Then you have to back off. (Don't stay close to him, so he can't hit you) His own Burst meter should be quite high, and he can execute you while getting hit, very annoying. So now that he is standing, press A and X. Keep spamming the A and X combo and he should be standing still getting hit once in a while. Keep repeating until his health is completely gone and you win the Match. Also a warning, ragequit. Expect it.
Mission 5: Win 10 consecutive matches via K.O.
You need to win 10 consecutive Matches in a row. You won't get all of your health back after each Match, only about 15%, so health is pretty limited. What you basically need to do is hit him with B spam at the beginning, use the Y attack, then spam B again and back off. Spam X and keep going backwards, so he can't hit you with his Special. That way you can defeat him from distance, and it worked like a charm for me.
Mission 6: No Mission
This is a strange one, with the first Lost Episode I unlocked the Achievement while I didn't even complete the Mission, so maybe it also applies for this one. If it doesn't, all you need to do is complete the Story.
If this Guide didn't help you, please let me know that you did to pass the Mission in a comment, so hopefully I can edit my Solution to help others!