Buying Frozen Synapse, either on Steam or from the developer's web site, gets you a key code for the DLC.
No matter how you download the DLC, you must activate it by running Frozen Synapse and pressing the "Activate DLC" button to paste in the code.
In addition, you can take the FS Red DLC code and activate it in Steam by going to Games->Activate A Product On Steam.
Whether you buy through Steam or through the developer's web site, you get a code that can be entered in FS and also added to Steam.
If you purchase from Mode 7's web site, the developers get more of the money!
From IRC (Omroth is Ian Hardingham, the lead developer):
(23:23:05) Goateh: Omroth! does it make a difference if i buy the dlc
through steam or through your site?
(23:23:54) Omroth: more money via us
(23:24:00) Omroth: and you can activate it on steam :P