Greek character is beginning to perk up (3)
I believe the answer is:
psi
'greek character' is the definition.
(letter from the Greek alphabet)
'beginning to perk up' is the wordplay.
'beginning to' indicates taking the first letters.
'up' becomes 'si' (I can't justify this - if you can you should believe this answer much more).
The first letter of 'perk' is 'p'.
'p'+'si'='PSI'
'is' acts as a link.
(Other definitions for psi that I've seen before include "letter from abroad" , "Greek letter; pressure unit" , "Pounds per sq. in. (abbr.)" , "Greek letter; pounds per square inch (abbrev.)" , "one employed by Socrates?" .)