Specific speech of king in classical garb from the east (5)
I believe the answer is:
argot
'specific speech' is the definition.
(language or vocabulary of a specific group)
'king in classical garb from the east' is the wordplay.
'king' becomes 'R' (abbreviation for rex, king in Latin).
'in' is an insertion indicator.
'classical garb' becomes 'toga' (ancient type of garment).
'from the east' says the letters should be written in reverse (on a map from right to left).
'r' going within 'toga' is 'togra'.
'togra' back-to-front is 'ARGOT'.
'of' acts as a link.
(Other definitions for argot that I've seen before include "Jargon or slang (of a group)" , "manner of speaking" , "Slang of a particular group, originally thieves" , "Groat (anag)" , "Particular group's jargon" .)