Yep. I'm breaking my silence.
In order to follow along with this tutorial, you have to have Photoshop. You can't get this effect on Paint. GIMP I'm not sure about, never used it. I haven't explored Paintshop Pro enough to know if it'll work on there.

Open up your image. If you want it bigger for a wallpaper, hit Ctrl+T (Transform) Drag the corner boxes while holding down Shift. When you get your desired size, hit Enter.
You are going to get friendly with the circled icons on your layer tray.
Blue: Create a new layer. You are going to be using this a lot.
Red: Create a layer set. This will organize all your layers.
Green: Lock transparency. Any transparent parts of the image will not be edited.

Select the pen tool. (P on your keyboard) Make sure your setup is the same as mine in the blue circle.

Zoom in (Magnifying Glass icon, just like on Paint. and click on the border of the image. You'll see a little dot where you clicked. Now click the end of the curve like so.

Wait, what? It doesn't curve like the ear! It will soon. Click on the line you just made. You have created an anchor point. Now hold down Ctrl and drag the point until you get a curve like the ear.
Now do this for the entire image! Yeah. The entire image. Don't worry if you don't get it perfect. The original screenshot won't be there when we're done. (Pay attention to the clock on step 4.)
When you get to the end of your path, your cursor will be the pen but have a little circle. Click when you see this and you have a finished path!

Set up your layerspace. Make a Layerset and name it whatever character you're rendering (I'm just using Balto, so I named it Balto.) And make a layer in the layer set called "Base." This will be your base color.
Select your base color. Use the most common color on the character i.e Steele is mostly white so I would use a white base color.
Because I'm lazy, I'm using the colors from my wallpaper "Over the Hills and Far Away" Once you have your color, select the "Base" layer, right click inside your path and select "Fill Path." Once filled, right click again and select "Delete Path."

Remember the Lock Transparency button? Click that to lock your Base layer. Then hit the eye icon on the left side of the layer tray. OH NOES IT DISAPPEARED. Don't worry, you just hid it so you can focus on the markings.
Do the same thing with the paths as you did with for the base. Except with your other colors.
There are some secrets I will still keep to myself (how I get perfectly round irises and pupils for one) but this is the basics of vectoring in Photoshop!

This is my finished result! I used brushes from
http://www.brushking.eu for the background. Overall, it took me about 25 ,minutes because this is a simple screenshot. I took a nap in the middle of this so it looks like I took almost 6 hours x3
On average, one vector render takes around half an hour. The longest I've spent on a vector render is 6-7 hours (Wolf's Rain Balto) and the shortest amount of time I spent on a vector render was 25 minutes (this tutorial.) Hey! New record!
Please let me know if I can clarify anything for you. I'm really bad at tutorials. Also let me know if there is anything in my gallery you want explained i.e. how I made a certain sig.
VECTOR FAQ
Q: Do you use a mouse or a tablet?
A: My hand is too shaky for a tablet (side effect of medication) and using a pen hurts my wrist after a while. So I use a mouse. I would imagine using a tablet for this would be a nightmare.
Q: Can I use your renders for colors?
A: No. It's not that hard to select colors from the original anyway. (You can get a rough palette by copying your layer with Ctrl+J and going to Filter->Artistic->Cutout.)