You really don't need to remove the white in that image.
All you need to do is change the Layer Blending Mode to Multiply. Since it's a black and white layer, using
Multiply will merely make the black visible and hide the white whenever it is on top of a darker color.
If you still want to go to the trouble of removing the white....
(Window > Channels) : Copy the
Blue Channel in the Channels Panel. Use
Image > Adjustments > Levels to increase the contrast of the black on the channel.
Blue Channel copy to load it as a selection.
Highlight the image Layer in the Layers Panel and then Option/Alt-click the
New Maskicon at the bottom of the Layers Panel to create a "hide all" layer mask.
I would still set the blend mode to Multiply for the layer. Then possibly duplicate the layer to further reinforce the black.
Then you can just paint or whatever on any layer below it.....
Channels can be a very powerful way of creating selections. You essentially just have to look for a high-contrast channel, remembering black means selected and white means not selected, similar to a layer mask.