Can I sell my characters on the Auction House?

I know that it's possible to sell gold, items, and consumables on the various Auction Houses within Diablo 3.

I heard some buzz over twitter that Blizzard was looking into also allowing characters to be sold. Is it yet possible to sell entire characters, for instance, if someone wanted to avoid the arduous leveling process?

While this is not possible now, in the future Blizzard is actually going to allow it. In their official Auction House FAQ, they state:

What items can be traded in Diablo III?

Nearly everything that drops on the ground, including gold, can be traded with other players directly or through the auction house system. Aside from certain quest items, there will be very few (if any) items that will be “soulbound” to your character and therefore untradable. We are also planning to allow players to buy and sell characters in the auction house at some point in the future and will have more details to share on that at a later date.

Source: Auction House FAQ

This is not possible yet, and they can always change their mind, but once they prove that the RMAH works, I would bet they go through with this. I'm sure they'll be happy to let you buy and sell characters at whatever prices you wish, and humbly charge a barely-noticeable percentage fee for helping you out.

No. No you can't. You can only sell items and gold.

Characters are not transferrable, the only way to sell them would be to sell your Battle.Net account, a practice that is in violation of Blizzards EULA and highly frowned upon by the community at large.