American Indian disposes of horse quickly (5)
I believe the answer is:
apace
'quickly' is the definition.
('apace' can be a synonym of 'quickly')
'american indian disposes of horse' is the wordplay.
'american indian' becomes 'apache' (I've seen this before).
'disposes of' indicates named letters should be taken away.
'horse' becomes 'h' ('H' can be a synonym of 'horse').
'apache' with 'h' removed is 'APACE'.
(Other definitions for apace that I've seen before include "At a rapid rate" , "flat out" , "fast" , "What's quickly" , "Swiftly, quickly" .)