First it is best to know what size (l+w, mb) avatar the website will allow otherwise your final choice may be in vain
and you don't have to stick with a still you use a GIF if you are permitted; some websites won't tolerate it or even has a list things you must do before your allowed
My question to you is what do you want? If you do not know, your search may be a bit longer
it is best to see all options in this tutional as you may change your mind how you go about finding one

1-5 Explain Different Option You Can Choose
6 How To Make A GIF
7 Editing Your Pics after Scanning or Photographing them in

Step 1: Use Websites Supply

true sometimes this isn't even worth trying when they haven't even sorted them out into categories and just have pages and pages of random picture but most sites do, thankfully

Step 2: Finding Within the Web

Use a internet search system ; search for stuff you like or think you may want which is basically stuff your interested in
some websites even make avatars to download like comic sites such as vgcats.com
don't worry if the image is too big as you can always resize it in photoshop or the free download gimp with the transform tool

Step 3: Take a Photo

Use yourself
could you be your own answer? The face that looks back at you in mirror? If it is why not have a picture of you dressed up as a character, just a pic of a fun day out or maybe just a randomly taken photo? But still be warned, using a pic of yourself may not be the best option and may be even dangerous to your well being but it's always a choice
you could always take pictures of other things like:
an animal- Family Pet
-wildlife
- zoo
-farm
Plants
Countryside
Beaches
it could be anything like places and objects

Step 4: Make Your Own

why not create an image by drawing something in paint or photoshop (or some similar programmes),

or do something off the computer by drawing an picture; you could color it in after scanning it onto your computer or downloading it after taking a picture from your digital camera

-how about an arrangement of downloaded pics spliced together

you can just give your chosen image some personal touches too make it more individual to you possibly something like a bit a text or a boarder

if you up to it you could even create a GIF yourself

Step 5: Ask Someone Else

Ask someone to make one for you either request of an online or offline friend or someone who is just so damn good you can't help but ask. The worst what could happen is you can get refused but its always something to try

Step 6: Making a GIF (photoshop)

look at pics for reference

draw one image then create a new layer and draw on the new layer with a slight difference to the first
if you see my second image the eye is squinted while the first is full
do layer after layer in till your done (best keep it simple for your first try)
as you can see my last picture the eye is completely shut

now go: File > Edit in ImageReady

you should have an box called animation now
Duplicate in till you have the same amount of frames you do layers

okay now click on the first frame and turn veiw off all layers except the first
then go on the second frame and turn the next layer on and the one previous off do this till you have finished them all

now play it
is it too fast or too slow just click the first frame then hold shift while clicking on the last frame then click on where it says 0 seconds then of coaurse change it how you wish
i'm putting mine on 0.5 secs so it goes slow to easily show the transition to the next frame

then finally when you finished
go to: File > Save Optimized As
give it a name ending with .GIF
and there you have it your very own moving picture

Step 7: Editing Pictures

~remember to save as you go~

Alright first thing to do is draw a picture anything you like or maybe just get a picture you already did

now scan or take a photo of it (don't worry if it's blurred or anything as long as you can see the lines of your drawing it is okay)
download the picture from your camera or save your scanned image

open it up in application such as Gimp or Photoshop

create a new layer and select onto it call it 'outline'
now draw around the outline of your picture
if you want to you can try improve along the way
though you probably already noticed i have in mine in (i also am pointing them out with arrows *shrugs*);
Blue Arrow - i didn't include the lips and nose as they don't look right so i'll add them in later
Black Arrow - i fattened up the arms
Green Arrow - adjusted her er clevige as looked too side drawn and didn't resemble the positioning of her actual posture
Purple Arrow - i removed unnecessary lines and tried to reshape the hand
now turn the background layer off to check that you copied all the lines

after that create a new layer called skin (or you could just stick to the fill and paint tools and be done when your done but it kinda easier this way as you don't inflict the other parts of the picture)
select onto and just paint the skin color wherever there is exposed flesh don't worry if you go out of the lines you can sort it out later
now select the sponge tool and set it to saturate you can change the flow and size of your brush to adjust to you as it may be too fat or too slow for you

remember patience is virtue
and if it doesn't look quite right don't just do undo work on other things to see how it works with the rest of the picture. You may find it actually works.

anyway sponge in places you expect shade and all around the outline but make sure the shaded parts are darker or thicker (hopefully my focusing on the stomach will help you understand more)
if you play around with it you may get a real understanding of shading

now create another layer
of whatever is in front of the skin of the skin (a single item mind)
keep making layers till you have everything in your picture singled out but you can still group together some things eg the pair of gloves, googles (with it's lenses), skin (don't seperate into each individual thing it is; ears, face, arms etc. unless you have a reason to do so)
think about what should be ontop of what
to help give a better idea on why it's so important how you layer them i will give you a list of mine:

Outline ->Must always be on top
Hair ->
Goggles -> must be under hair and on top of skin at least
Lips -> doesn't matter as long as it's on top of skin
Eyes -> doesn't matter as long as it's on top of skin
Scarf -> doesn't matter as long as it's on top of skin
Tank top -> must be under scarf and on top of skin
Right Pocket -> over gloves to cover them
Gloves -> over jeans, belt pants and skin. under right pocket
Buckle ->
Jean ->
Belt ->must be under buckle and jeans (don't color where it is)
Pants -> under everything except skin
Skin ->guess it must be under everything
Original -> Can be kept but might as well be deleted

sorry bout that (could you fill in the blank spacesanyway you should be able too complete your picture with some of techniques i taught you (it's kind of the one technique huh)
when it comes to the hair think about where the light will shine and use desaturate as well for that

well i hope this has been at least some help as do seem to complicated things and kind of rushed through everything but i hope i have been of some help
well see ya
oh wait i forgot please show me how you got on with it, it'll will be lovely to see what you created