The most common and most likely answer to this crossword clue is the 5 letter word PIOUS
I've seen this crossword clue in The Times
P
I
O
U
S
More clues leading to the result PIOUS
- Leader of prayers has obligations for devout
- Religious in abundance, but dismissing heads of Catholic order
- Religious minister not coming in earlier
- Self-righteous Liberal short of company
- Religious leader in prison over debts
- Further thoughts about obligation to be devout
- Having second thoughts about obligation to be devout
- Devout priest promises to pay
- Constant love: our lot devoted to God
- ‘Dutiful’, suggesting French ranker?
Buddy explains!

The straight part
We know that "Self-righteous" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case self-righteous = pious.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does PIOUS mean?
PIOUS adjective- Of or pertaining to piety, exhibiting piety, devout, godfearing.