This is disputed. It depends on one's definition of being a Time Lord/Lady. River Song's parents were humans, so she is not a Time Lord by descent. However, she has some of the qualities of a Time Lord, such as regeneration. Also, examination of her DNA as a child revealed that it was similar to that of a Time Lord.
Many sources say that the Gallifreyans became Time Lords because of prolonged exposure to the Time Vortex, but this is disputed as well. If this is the case, than River acquired Time Lord characteristics the same way as the "true" Time Lords did.
The Silence may have also modified her in some way to make her closer to a Time Lord, to use as a weapon against the Doctor.