This is a problem i deal with a lot when creating images used to key over other images.
Key or matte effects require that the background of the image that is keyed over another image be a suitable, solid continuous color.
In order to achieve this effect, I use a "loud" color like blue or green to fill the background. this "loud" fill color exposes pixels that you missed when trying to create a uniform background color. Simply use an eraser tool in your favorite graphics editor (mine, perversely, is Windows Paint) until you have a solid background with no artifacts - like the thin, ghostly border lines in your graphic, above.
Once you have completely cleaned up the background, you can use your fill tool to change the background color to whatever you want - like white in your case.
Here is the really important part: DO NOT SAVE THE GRAPHIC USING THE GIF FORMAT! Doing so will re-introduce off-color pixels into the background leaving you back where you started - and possibly worse. Save the cleaned up graphic in a file format like 24-bit .bmp that saves solid colors as solid colors. Then, and only then, can you experiment with useful graphic formats like JPG or PNG to see if the key-able/matte-able solid background color that you wanted remains that way - solid.
I hope that this helps!
