This Adventure mode is composed of 60 stages as opposed to the 55 from the original game. The first 50 stages follow a similar formula in which you have to beat a sequence of five levels with a specific character as follows:
1) Bjorn Unicorn (Super Guide)
2) Jimmy Lightning (Multiball)
3) Kat Tut (Pyramid)
4) Splork (Space Blast)
5) Claude (Flippers)
6) Renfield (Spooky Ball)
7) Tula (Flower Power)
8) Warren (Lucky Spin)
9) Lord Cinderbottom (Fireball)
10) Master Hu (Zen Ball)
Stages 11-1 through 11-5 will require you to use the newly added character, Marina (Electrobolt).
Stages 12-1 through 12-5 are like the last 5 levels in the previous adventure mode in which you can pick whichever character you desire to complete them.
There are many ways to go about these last 5 levels, but Warren or Master Hu are highly recommended. Marina is also a good choice depending on where the orange pegs are placed and if you can hit a green peg early in the level. If none of these are working for you, then these are the ones I'd recommend for each level:
12-3: Marina (she's by far the most successful for this)
Keep in mind that you can retry any level as many times as you like, but you won't be able to advance forward without completing each level.
Once you finish 12-5 your achievement will unlock and you'll get a swanky Trophy to boot. From here you can retry the Adventure Mode with your choice of character for each level although there are no more achievements in it for you.