Why do cargo airlines frequently choose passenger aircraft rather than aircraft designed specifically for cargo?
Price, commonality, and size choice.

A very large proportion of an aircraft's cost is in design and certification. It's followed by the engines, the avionics, and the fuselage. Not much cost can be removed from an aircraft by removing the passenger-specific elements of design. That is done in freighter versions.

Freighter models like the 747-8F already realize most of the possible savings. Second-hand airliners are cheaper still, and they're not hard to convert. Rip out the interior, install cargo securing systems, and fly. 

Commonality with airliners means airliner-derived freighters are easier to service and repair anywhere one needs to. And, as Dohn Joe rightly adds, get pilots for; cargo airlines are both a training ground for passenger pilots and a career path of its own.

Finally, there are more airliner models than freight-only aircraft models, so there's more choice of the right payload and range capability. 

This reuse of passenger airliners for freight is a great example of efficiency. Passenger airlines tend to have very high utilization, putting lots of cycles on their airframes and lots of fuel through them, so they care for better fuel efficiency, availability, and passenger experience (or at least their expectations when booking, considering the 9-abreast 787) delivered by newer airframes.   

Cargo airlines run looser schedules, because their clients don't care much for their flights leaving on time (air freight is usually offered in 2-5 days), which shifts the optimization focus from fuel efficiency towards capital costs. They are able to use up the rest of the airframe's life in a less-critical job, where the oversight is a bit more forgiving and delays due to lower availability are more acceptable.

The An-225 is special in that it occupies a class above every other freighter in capability. It does the jobs nothing else can, but, until recently, the only airframe built has been able to serve the whole world.