The example you posted shows the image has no transparency. This can be seen in the layers panel. Adding a transparent layer underneath your image layer will not create any transparency. Save for Web won't fix it either. Instead, you need to add a mask to your drawing to mask out the white background to make it transparent.
There are several ways this could be done. You can try this very basic method which will work for most black and white drawings:
Open the Channels panel, Ctrl+click on the RGB channel to load it as a selection. If the image is in Grayscale mode, do it on the Gray channel - there will only be one channel anyway. If in CMYK mode do it on the CMYK channel.
Do Shift+Ctrl+I to Invert the selection (or Select > Invert)
Go back into the Layers panel, make sure the layer is selected by clicking on it, and hit the Layer Mask icon at the bottom
Do Ctrl+D to deselect everything (or Select > Deselect)
Example

- Sometimes this will leave little white pixels around the edge. You can fix that easily by selecting the image thumbnail in the layers panel. Then do Edit > Fill and choose Black as the fill colour.

Now you should be able to export as PNG with transparency