How to cut up a sprite sheet and make the background transparent

Download GIMP. Open the picture in GIMP, click on the "color" drop down menu at the top of the GIMP window. Select "color to alpha". A tool option window will open. Click on the eye dropper icon, when your cursor turns into a square click on the background color. The background color will disappear but the images you want will become partially transparent. Play with the "opacity" slider until the images you want are no longer transparent. Close the "color to alpha" window.

Now you have your sprite sheet with transparency. Next, if you want to turn them into individual sprites, copy and paste the individual frames into new pictures, select transparent background in the advanced settings drop down menu when you do. 

After you copy and paste them, click on the "image" drop down menu and select "crop to image". This will shirk the picture to the image size.

Finally, go to the file drop down menu and select "export as" and turn them into PNG files. 

Transparency depends on how you're doing it - via PNG or the format that Doom uses for sprites. If the former, you'll set the cyan to be transparent in some sort of image editor (i.e; Photoshop, GIMP, etc.) If the latter, you can set that in SLADE.

As for cutting it up, basically cut them broadly into sprites, then trim down the sprites (most image editors will have a tool that will strip away unnecessary extra space on the edges).

Either way, you'd need to align the sprites in something like SLADE for it to look good.

Both of the above methods are valid.

I will add my third method because

a) it's good for helping you think about what you are doing

b) it's already written up in response to another who was in your predicament.

http://chapsoftware.com/spriteColoring.htm

Awesome, thanks for all the help guys. I was able to compile the sheet into their sprites.