This achievement is for completing five missions in a row without failing any of them (as failing a mission restarts the count).
As such, I recommend doing this achievement in a group, as your group members can help with mission objectives and save you if caught by the shape/sonic barriers/etc. I found you are much more likely to fail a mission running solo than in a group, and the more people the better. Max out at four if you can.
Edit (July 4, 2021): According to TuKraZe: the fivemissions no not need to be completed in one session, and the achievement does not need to be done with the same players. More good news: you can lose connection between matches at any point and not void your progress.