Generally, Kiev is safe. Most apartments have pretty solid doors and, provided you remember to lock them, there's no need to worry overly about laptops. You could be an unlucky victim of a burglary, as anywhere, but Kiev is not especially bad.
My wife confidently wlaks around central areas of Kiev late at night, never had problems. We're also from Surrey, UK BTW.
You will notice a more conspicuous mafia presence than in most western cities. Usual mafia stuff - protection rackets, prostitution, drugs etc. Dangerous to get involved. If you see a 200kg gorilla covered in bling with surgically enhanced peroxide blonde girlfried, don't try to hit on the girl.
Summary - there's trouble to find if you go looking for it, but it doesn't have to cause you trouble as a visitor.

