Here is the full list of all 10 Shin Samurai Jazz achievements.
Finish the first act...
Finish the second act...
Finish the third act...
Finish the fourth act...
Finish the fifth act...
Finish the first act with less than ten deaths...
Finish the second act with less than ten deaths...
Finish the third act with less than ten deaths...
Finish the fourth act with less than ten deaths...
Finish the fifth act with less than ten deaths...