The main problem in such an icon is the antialiasing zone between the black and the white.
@ybeltukov approach with GIMP is correct. In order to proceed and clear the white zone you need to select it (by color or with the magic wand) and tweak a little e.g. increasing the selection by 1 pixel and feather the selection to avoid the pixeling effect visible in your example.
In Inkscape is not so difficult.
At first you need to import the icon, while dragging it or using File->Import... (you can either embed or link the icon, for this example it doesn't matter).
Next step is vectorize the image, i.e. convert the pixels of the image in vertex and lines. This task could be performed using Path -> Trace Bitmap... (the icon must be selected). It appears a window with the settings:
You need to find the optimal settings for your image. With "Update" button you can have a preview.
When done, hit "Ok" button and close the window. The traced image is over the original one, as a group of paths.
Now, you can ungroup the image, choose the paths more close to your icon, edit a bit them (refer to a manual) and change the colors as you need:
Now, you can export the image (File -> Export Bitmap...) as a png (note that the inner of the image is transparent, now):
So... using GIMP or Inkscape? Depends of your needs.