Dirty sort of business a Republican goes after (5)
I believe the answer is:
smear
'dirty' is the definition.
(smearing is a kind of dirtying)
'sort of business a republican goes after' is the wordplay.
'sort of business' becomes 'sme' (I can't explain this - if you can you should believe this answer much more).
'republican' becomes 'R' (abbreviation).
'goes after' is a charade indicator (letters next to each other).
'sme'+'a'+'r'='SMEAR'
Can you help me to learn more?
(Other definitions for smear that I've seen before include "Dirty mark; a libel" , "False report" , "allegation" , "Greasy/sticky mark" , "nasty mark" .)