Croon wildly, having married one foreign character (7)
I believe the answer is:
omicron
'foreign character' is the definition.
(Greek letter)
'croon wildly having married one' is the wordplay.
'wildly' indicates anagramming the letters.
'having' indicates putting letters inside.
'married' becomes 'm' (abbreviation, e.g. in genealogy).
'one' becomes 'i' (Roman numeral).
'croon' is an anagram of 'ocron'.
'm'+'i'='mi'
'ocron' placed around 'mi' is 'OMICRON'.
(Other definitions for omicron that I've seen before include "character in Sophocles" , "Foreign letter -- I'm no roc (anag)" , "Greek character" , "Round character" , "letter from 13 [HELLENE]" .)