The first time you run any published scenario, one of your characters gets credit, whether stand-alone or multi-part. And as far as I can tell from what has been posted elsewhere, the only time it matters is if you are playing them. I am pretty sure that, for example, if you run a four part scenario arc, you could give four different characters each a chronicle sheet.
Now this next part Joshua may need to clarify, but it is my understanding that while the parts are to be played in a certain order, you do not have to use the same character in every part. Otherwise, again for example, if you had a character killed in part two of a four part arc and you had no way to revive him, you would be screwed out of played the final two parts, unless in the extreme rare case of replay you were able to play the first two parts again and then got to play parts three and four with the new character as well.