Use the magic wand (you can vary the tolerance) and select the white, then do whatever to the selected area.
Or if you want to change the white to a solid colour, just use the fill bucket (maybe hiding under the gradient tool on your toolbar) and again select your tolerance and fill with your foreground colour.
OR the way I would do it..
Use the magic wand (you can vary the tolerance) and select the white, the go to select inverse. Then copy and paste (automatically to another layer).
Then do as you wish to the layer underneath the logo layer. That way you can put your logo on a background that's not solid.