In Quickplay, when selecting a setlist, you have the option to tag a mix as 'Liked' using the yellow button.
To get this achievement you need to tag 10 mixes as 'Like' and then play them. I accomplished this simply by liking them then playing them straight away in quick play, but it appears that plays in other game modes (e.g. Empire Mode) also count.
For simplicity, you could do this at the same time as you go for the 'Quickathlon' achievement, by liking 10 mixes, then creating a setlist with those 10 mixes and playing it (earning at least 4 stars on each song).