I'm wondering how leveling a character works from base set to base set. For example: Say I have played Merisiel through Rise of the Runelords adventure deck 6, now when I start Skull & Shackles Base Set does she keep her stats?
You start a new character when you start an adventure path. That can be the exact same version of Merisiel that you played before, or it can be the Skull and Shackles version of Merisiel. Or it can be any blank, new character.
There is no "leveling" or "keeping her stats" from base set to base set.
Your character at the end of a base set is way too powerful to have any fun playing a new base set.
As Hawk said, you have to start back with a blank character sheet (no slills, no powers, 15 cards) and a deck with only basic cards (and I recommand getting them at random, so you'll have more opportunities/fun to loot/improve).
But you can indeed restart with the same character, a new version of that character (in this case I agree that if the base set you start has a version of your previous character, it's tempting to go for it), or just change altogether.
Well you can take those conquering characters into deck 4 of Wrath of the Righteous. How to do so is laid out in this blog.
http://paizo.com/paizo/blog/v5748dyo5lhxk
But if it's home play it's really up to you what you want to do with characters who have completed sets. Paizo really seems to encourage you to house rule things to be fun.
Personally after completing a whole path with a character I kind of want to try out a new one, but I get that Rise & Skulls can leave characters feeling incomplete 'now what?" especially given the giant pile of rewards and 2x card feats when you are DONE...
If you WANT to take a character that completed RotRL through S&S (or vise versa) and stack the rewards and your group is okay with being overpowered that's up to you.
There are a few house rules you could run with that would make it less OP:
1. Treat ALL banes in the new set as "Veterans" with "Increase the difficulty to defeat by the adventure deck number." and start off counting deck 1 of the new set as Deck 7. (You may have to scale this back rather than just keep adding + each deck, "AD12 veteran Banes" might be statistically too high, since a character who completed a Path is going to grow broadly in power, rather continue to go straight up in power. Chances are you are already able to make close to the largest possible rolls your character can produce in your main skills by the time you complete one path. extra feats are probably going to make you better at the stuff you are weak in & not much stronger at the stuff you are already good at.)
2.Just don't use your Role Card in the new set until you would earn it at the end of deck three. And treat all your skill checkboxes as capped by the current Adventure deck number (i.e max +1 in deck 1, max +2 in deck 2). Treat all your boons as if their AD# were only as high as the AD you are actually in.