Language, almost dead, cool (5)
I believe the answer is:
latin
'language' is the definition.
(I know that Latin is a dead language)
'almost dead cool' is the wordplay.
'almost' means to remove the last letter (most of the word but not all of it).
'dead' becomes 'late' (similar in meaning).
'cool' becomes 'in' ('in' can mean fashionable or cool).
'late' with its last letter taken away is 'lat'.
'lat'+'in'='LATIN'
(Other definitions for latin that I've seen before include "ancient tongue" , "Language of Cicero" , "Language of old Romans" , "demanding subject" , "A Romance language" .)