Q:
Achievement progression account-bound?
A:
Achievements are for the account, not character specific, so you can use alts to work towards the total kills etc. The wiki also mentions using alts in the notes part.
https://wiki.guildwars2.com/wiki/Guide_to_Slayer_achievements
You can use as many characters as you like, it’ll all contribute to your progress.
As others have said, Achievements are account wide. Some achievements require you to do a series of things on one character, such as the World Completion achievement, but they are a vast minority.
Thanks, everyone!