There are a few reasons to have a bank alt. First, bag and bank space. Since you don't want to "flood the market" with any given particular item (because that drives prices down), you might need to put some items in cold storage until the market stabilizes. While your main character's bankspace might be gargantuan, most people find the additional bag and bank space of a bank alt attractive.
Second, if you're a member of a social guild, having a bank alt lets you focus briefly on the Auction game. Many players check their auction mail before they go to work, and don't want the awkward social action of ignoring their fellow guild peers when only on for a very short time.
Third, there's an inalienable style to the bank alt. Even now, WoW.com is cruising for submissions for the most stylish, awesome bank alts out there. It's a subculture in WoW. As Penny Arcade once so brilliantly illustrated: "Naked they run, level 1 banks: Clark Kents of Azeroth."
The first step in creating your bank alt is its name. While any name works for many people, I find that the discriminating bank alt requires a certain flair to their name. You're going to be listing dozens -- hundreds! thousands! -- of auctions, and the in-game market will dance at your command. Names like "warriorlol" or "funkimage" just don't get the same feel across to your customers. Your bank alt doesn't need to sport a roleplay name, necessarily, but the moniker should be memorable.
In order to find inspiration for your bank alt's name, I'd suggest looking to history first. Perhaps you can find a famous or notable banker to serve as your alt's namesake. Morgan, Chase, and Lehman all seem like solid starters. You could also find a good referential name -- perhaps a Latin word for money, or bank, or some other concept.
Personal preference has a lot to do with which class you should choose for your bank alt. However, if your bank alt's going to stay stylish, then I would highly encourage you to choose a plate-wearing class. Because death knights, warriors, and paladins can wear any given armor in the game, your bank alt would always be able to sport the latest low-level fashion available.
Choosing between death knights and warriors for your bank alt can be very difficult. Death knights start off the game with the highest level and all of the old world flight points. As well, once you get your death knight out of its starting area, that character will be equipped with a mount and a teleport to Acherus. These are certainly useful skills. By comparison, the warrior class has the widest available weapon-wielding options (in case there's something you want to wield in particular for the sake of style), and death knights are awfully high level for a traditional bank alt.
The real test for which class you should choose as a bank alt, however, has to do with how high you'll be levelling your little money maker. If you're going to get your mage high enough to Blink, for example, then you might find that little hop-skip-jump ability saving a few seconds between the AH and the mailbox.
Most bank alts have a city of residence. Since bank alts tend to be lower level, they pick a city, and conduct their economic master plan from that one place. Of course, which city is the favored bank alt location depends on your faction.
If you're Alliance, the two predominant choices are either Stormwind or Ironforge. The big advantage in these two cities have to do with the very close proximity of the Auction House to the city's banks. You'll have to gauge which of the two cities have more loiterers hanging around the area. It's up to you if you want to avoid extra people to minimize your graphic lag, or if you'd like more folks around to show off your stylish bank alt.
The Horde have a single city whose convenience make it second to none. Thrall's very own homebase of Orgrimmar is, hands down, the best choice for a Horde bank alt. The Auction House and bank are very close to one another. There are usually just enough Horde members lingering about to show off without affecting your computer's performance. I've heard of strong arguments for Undercity, since it seems the undead keep an Auctioneer on every corner, but it feels like a longer run to the bank in the center.
Your bank alt probably won't be seeing the highest level of crafting skills, so there's really only a single choice for a bank alt's profession. That is Enchanting. This is because enchanters have the ability to tell what magic items can be disenchanted. Even if the alt can't do the item breaking itself, you'll always want to know that information before you drop coin down on an underpriced item.
The most important part of a successful bank alt is style. Zach has a wonderful guide to bank toon fashion which is still perfectly viable. I've seen a lot of people rocking out the Winter Clothes since the most recent Winter's Veil. The various seasonal clothing is a nice touch of class to many bank alts, since they provide a sense of longevity to a character which is (otherwise) not levelling.
The final piece to building your ideal bank alt is going to be a controversial choice for many of you: what to name your guild. Many fashionable bank alts choose to wear a vanity guild plate. That is another way of saying "a guild without members, which will keep you from getting random guild invites, and will also provide you additional bank space."
You'll have to convince some folks to sign your guild charter, which can be a little difficult in crankier servers. However, once you've got your guild charter signed, you're the proud owner of not only additional bank space on your character, but also a guild bank. This should provide ample room for you to grow your stocks.
The Final Word
If you're a successful auctioneer, you will eventually get someone who wants to talk to you. I'd suggest maintaining a professional customer service attitude when you deal with your fellows. Address your customers are "Sir" and "Ma'am," and be friendly and helpful. Your bank alt is a business person, not a silent tool. Feel free to enjoy these interactions -- they're the final token of your success.
Now get out there and get building those bank alts.