Give a man a fish...
Oh fine.
You can get
A gif animator/editorDownload that, And load your image into it.
Make a Temp folder, you will need this.
In Ulead gif animator, after you load your gif, go to Edit>Resize image.
Resize it to One of these sizes:
32x32, 64x64,126x126, 252x252.
The higher the size, the bigger the file, And the better the quality.
The file has to be less then 250KB to be loaded into TF2, or any other source game.
After that, Go to File>Save as>Image frames
Select your Temp folder, and save, This will create a picture for every frame in your gif.
Download
VTFEditOpen VTF edit, go to File>Import.
Go into your temp folder, and select all frames in the folder.
If you did it correctly, it should play the gif in VTF Edit.
File>Save as, name the VTF.
Now, copy this picture into:
C:\Program Files (x86)\Steam\steamapps\[Your steam]\team fortress 2\tf\materials\vgui\logos
And
C:\Program Files (x86)\Steam\steamapps\[Your steam]\team fortress 2\tf\materials\vgui\logos\UI
Launch tf2, go to that directory, find the file, and select it.
I think that's it.
Spaced it out so it's easier to read.