Cost of property starts from 1000gp for an apartment in the cheapest part of Old Korvosa (Old Dock), and goes up depending what you want to spend (up to around 20000gp for the most expensive apartments).
Moving into unclaimed property seems to be the tactic of 90% of the bad guys so far in this campaign - if a building is left without ownership due to bankruptcy or intestate, the City holds it in escrow until such time as it can be auctioned, or a legitimate claimant can be found. Since this is a process that can last indefinitely, people like Gaedren Lamm or Rolth tend to take advantage of this.