Social climber drinks headless bitter (7)
I believe the answer is:
upstart
'social climber' is the definition.
(thesaurus)
'headless bitter' is the wordplay.
'headless' becomes 'ups' (I can't justify this - if you can you should give a lot more credence to this answer).
'bitter' becomes 'tart' (I've seen this before).
'ups'+'tart'='UPSTART'
'drinks' acts as a link.
I am not very happy about this link. It may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for upstart that I've seen before include "Bumptious parvenu" , "Over-assertive newcomer" , "Bold newcomer" , "Nouveau-riche" , "Unexpectedly fresh" .)