All Avatars and Fonts

Buy an item with rarity 80 or higher at Unis Clothing.

Tip: Visit SunnyNeo.com's r80+ Clothing List to see all rarity 80+ clothes (with pictures!).

You get this avatar while playing Cellblock. Tournament level 3 seems to be the lowest level people have reported getting it thus far.
The Neopian Times Editorial Issue 210 said getting the avatar is not random, as you can see below:

TNT, can you make it where you can go back to tournament 1 on cellblock, most game avatars are hard to get, true. but dang this is just crazy, at least on flash games it dosnt just keep getting harder and harder to the point where you cant play it anymore, thats just crazy. it bugs about 30% of the people that play and arnt lucky couse you made the avatar random, i mean why does it have to be "SUPER HARD" AND "random". dosnt make it fair one bit for us unlucky ones.
We will look into a reset feature, however the avatar is not random. Not random at all. From what we've seen, not one person has actually figured out the real trick to getting the avatar. It may seem random, but it's not by any means. I think it's the only one we've ever really stumped people with. *cackle*

Score 900 or more points at Gourmet Club Bowls.

Tip: Type 'shepherd' after the game is loaded, after doing so, all points you score during the game will get doubled.
Tip: Type 'superbowl' to skip the current level you are on. No points will be rewarded for the level. Multiple use per game.

Beat the Devilpuss at Neoquest II.

Tip: The Devilpuss is located in the Cumulonimbus, Chapter 5.

Get beaten by a Plains Lupe at Neoquest II. You will get the avatar after you click the "Return to your last rest spot..." button.

Tip: Plain Lupes are located just outside the village you start the game at.

Beat Kasuki Lu in the 1 Player Battledome.

Tip: Visit SunnyNeo.com's Battledome Challengers page to see how you can get this challenger.

Send someone the Puppyblew "I Love You (Animated)" Neogreeting on Valentines Day (February 14th). This avatar returns every year, if you missed it, you can get it next year.

Be one of the first 250 people to win a Lenny Conundrum Contest.

Do one of Jhudora's Quests on Jhudora Day (February 4th). This avatar returns every year, if you missed it, you can get it next year.

Tip: Visit SunnyNeo.com's Jhudora's Cloud Guide.

Keep refreshing the Avatars FAQ page. Getting this avatar is random.

Visit The Beach while you have a Kacheek as your active pet.

Score 1,200 or more points at The Castle of Eliv Thade. To get those 1,200 points you have to keep on walking and solving anagrams untill you have enough points, just collecting the items isn't enough.

Tip: If you type in 'rehaxtint' after you pick easy or hard the game will start with one extra letter hint.

Score 1,000 or more points at Gadgadsgame.

At Grumpy Old King, tell a joke starting with the question: "What do you do if fierce peophins has eaten too much tin of olives", the answer doesn't matter. Getting this avatar is random, it might take a lot of tries before you get this avatar.

Have 1,000+ different items in your Safety Deposit Box.

Tip: Visit SunnyNeo.com's Packrat List to see a list of more than thousand of the cheapest items in Neopia.