Can I replace an existing receptacle with a Leviton USB receptacle?
At the link you provided, Leviton has a product sheet with installation instructions. As they note in there, if you need to connect two wires each for line and neutral, you must use the backwire connections:


  To Side Wire (if more than one wire is required on a terminal, you cannot side wire and must use the backwire option)


What that means is that instead of wrapping the stripped wire end around the screw, instead you insert a straight portion of stripped wire into holes in the back. Sometimes those are spring clamped just by inserting them, but here it sounds like the screw will tighten a clamp onto the inserted wires.  

You might be able to straighten your existing wires to do that, or cut and strip according to the instructions, and slide the pairs of wires under the clamps.