To unlock some of these archers, you'll be required to unlock the corresponding stage first.

Archer	How to unlock
(Cyan Archer) Prancing Puppet	Once you unlock the Sunken City stage.
(Purple Archer) Vicious Vessel	Play VS on the Twilight Spire stage until a random event occurs, may need to use the Turncloak Soldier/Last of the Order character to trigger event.
(White Archer) Ancient Exile	Once you unlock the Moonstage stage, play VS on it, and break the purple crystal that contains her, might have to play a few rounds to get the crystal.
(Yellow Archer) Forgotten Master	Collect all 7 yellow gems hidden throughout stages in Trial Mode.