When you are on the World Map screen, press LB to bring up your missions.
From here, press Y or Triangle to load your daily missions.
Complete ten of these and the achievements will unlock. You can complete missions two days in advance to speed up the time it takes. And you can take your pc offline, switch the date to literally anything, even the day after, load the game up, reconnect to the online and you will either have entirely new challenges for the dates in question, even the same day. All of them are randomly generated and they reset when you go offline.
Most missions are pretty easy requiring you to change your body parts or wispon, or complete a stage with a certain character or wispon.