Like the "10 Years of Service" achievement, this isn't quite as bad as it seems.
You can transfer mid-season, so from the moment you get signed as a youngster, to the moment you sign for your 10th club, you only need 4.5 seasons for the achievement to pop.
You also don't need to actually play any games it would seem, i.e you can choose the "rest" option for every game. It is worth considering though, that if you never play, therefore never score, never make any contribution, and as such improve your skills at a much slower rate, the offers from other teams may be very limited. This is especially important as you need to play in at least 4 countries. Offers from abroad may not come in unless your player has something to offer in terms of abilities or some personal titles to his name.
If you are purely going for the achievement, I would recommend playing for the more lowly teams, so you don't have to bother with European games or domestic cup ties past the first round or so. What I did was rest for the cup games, to better the chances of my team not progressing, and play only league matches, and only against the league's better teams. In most cases I was able to win the league, or at least get my team very highly placed, and be the top scorer and assist leader, while only playing between 3 and 7 games for each team. This means you get more choice when it comes to transfer offers.
Furthermore, if you really want to speed this up, it's worth considering the Dutch league as the main focus of your career. The league has only 18 teams, therefore 17 league games per half season rather than 19, and the cup games are one leg only. You only need to play in 4 different countries, so 7 Dutch teams and 3 from other leagues will do it.
Please note that, as far as I am aware, you cannot rest for internationals, (or otherwise declare yourself unavailable for selection), so if you are picked for your country, you'll have to play. Consider it a way of increasing your skill stats and winning some silverware.
Oddly, the achievement popped for me once I selected "go to the next week" after accepting an offer from my 9th club, not my 10th. It seems others have experienced this (thanks, Kamelbarn), so be aware. If this doesn't happen for you, just play one more season and you should have it.