No. The background actually has little to do with the logo itself. It’s the alpha channel that you need to modify. I’m assuming you’re using Photoshop based on the images you’ve provided, so here’s a brief explanation using one of the textures from the prop effect pack I’m working on for Ryu-Gi.
When you’re in Photoshop, there is a tab labeled as ‘Channels’ (which you can actually see in one of your provided screenshots in this thread. Going into that tab will normally show four separate channels; RGB, Red, Blue, and Green. The alpha channel, which usually isn’t present, needs to be made manually; you can do this by clicking on the ‘Create new layer/channel’ icon.
After clicking that, you should have a new channel appear in the list that’s labeled as ‘Alpha’ or ‘Alpha 1’. This channel will first appear as solid black or solid white; this is normal, and usually comes up as solid black (indicating that everything is hidden). From here, click on the RGB channel, and you should see your main work area return back to normal. Assuming you’re working on a single layer only, you’ll want to use one of the selection tools (marque tools, magic wand tools, etc.) to select only what you are wanting to have shown; since the logo is all one solid color against a black background, let’s just use the magic wand tool to select the colored segments only. After doing this, go back to your alpha channel by simply clicking on it under the Channels tab.
Since your logo doesn’t really have any defining edges or need for detail, you’ll either want to take your brush tool to the the selected areas (it will be white by default when you’re on the alpha channel) and simply go over the selection, or you can simply hit the delete key your keyboard to turn it white (if you know how to flip the color swatches in Photoshop, you may have to do so in order to have your selection turn white while in the alpha channel; hitting the delete key by default without swapping the color swatches will leave it black if you don’t, so it may be necessary to flip them around). After doing this, save your .TGA and make sure it retains the alpha channel. Once you’ve converted it over to a .VTF, the alpha channel should remain. All that would be left after that would be to use either $translucent 1 or $alphatest 1 in your .VMT (preferably $alphatest since it gives you a bit more freedom to work with other commands) to have it use the transparency provided in the alpha channel.
On a side note, it would actually be wise to have two separate .VMT files that make use of $translucent and $alphatest separately, as well as to make a second layer of that logo in your 3D modelling program and assign both layers one of the two .VMT files during the compiling process. By doing this, you can make the logo blend in a bit better, and you can adjust the .VMT with $alphatest included until you get the desired results for the logo.