While the achievement is pretty self explanatory, it's not going to be very quick. Just to give a very rough time estimate, the title update landed early December and I unlocked this January 2nd. I wouldn't say I am the strongest or most dedicated player, but I had the components necessary for
10 composes within a month. So what goes into this? Here is the gist of it all:
Within the dungeon, there are treasure chests (approximately 3 chests per 10 floors). These chest contain scraps to compose new cards (mostly 4 and 5 star cards). Additionally, it is possible to find universal 3/4/5 star card scraps. With enough scraps (and gold), you are able to eventually create the cards from those scraps.
Every card you compose can be 50% Universal scraps while the other 50% (at minimum) must be for the actual card. In general, the universal scraps are much rarer and very versatile. In general, most 5 star cards are roughly 80 scraps total, while 4 star cards are 120. These numbers do vary however and can be higher for 5* and lower for the 4*.
While these universal scraps can help you get a card sooner, I'm going to recommend saving the universal scraps. Since Title Update 1 came out, Title Update 2 has been released which introduced the Evolve and Shuffle mechanic. The Universal 3/4/5 Star Cards can act as Evolve and Shuffle material for cards of the matching star level. With how difficult the later maps (11 and 12) and Dungeon floors are (anything above 50 really), it is going to be highly important to have decent evolved cards with high level skills.
Therefore, continue to save up until you can completely compose a card with it's own scrap types. After about a month, you should have enough scraps for the composes (or be very close at least). In terms of the dungeon itself, you should look to at least get through Floor 50 everyday. This gives you a decent amount towards the achievements and the scraps for the achievements. It will still take a long time, but in all seriousness, if you're going for this achievement, then you're already in for the long haul and grinds in general as you've been doing it forever already
.