Say I have Cloak of the Martyr equipped to a figure. Say I have 3 spellcasters perform spells within the range of the Cloak wielder, would I be able to give the wielder a click for each action taken? Could I even kill (martyr, hah!) a figure in a single turn by doing this?

I would say yes, unless the text of the Cloak of the Martyr says something about only being used once per round.

The card does not say you can do it only once per turn so you definitely can push the wielder multiple times per turn. That's why I usually have it equiped to a figure with Regeneration (Aovon). Figures with the Rally subfaction ability may be good candidates as well, though I'm not sure of any viable ones.