Ząbkowska Street used to be pretty bad, esp. the tunnel-like doorways leading to the court yard. There used to be a distillery there. Is it sitll there?
Zabkowska has been cleaned up at great cost, but the warrens of dodgy courtyards still exist in that area (and by the way, the distillery is still there). One thing that's happening is that district hot water is being rolled out round there, paid for by the EU.
If you want slums, you'll often find them cheek by jowl with posher apartments dotted around Srodmiescie. Communal flats that people aren't paying for and in buildings that aren't looked after.
Someone mentioned Wola. There are some dodgy bits between Mlynarska and Wolska however I wouldn't call them slums - the housing stock is actually quite good.
There are some twentieth century housing estates dotted round town that are pretty bad - again, very scruffy communal tenants and shared bathrooms. There are some buildings like that even at the bottom of Zoliborz Oficerskie that resist gentrification because of the neighbours.