It's not really brewing, but the best way to make one of these sorts of beverages is as follows:
1) Purchase inexpensive vodka
2) Dilute to the appropriate % alcohol
3) Add desired flavoring
4) Force carbonate
To actually 'brew' something like this is, as Triple Freak mentioned, far too involved to be worth the effort. You would need to invest in some serious filtering equipment to do a decent job of removing all of the malt character from the beer. The breweries only do this to get around the extra taxes on distilled spirits.
One thing you might look into would be an alcoholic ginger beer. That would give you the soda-like quality in a brewed beverage, though the recipes I recall didn't have any barley malt in them. You could also give mead a try. It will have a lot more character than Smirnoff's, but it might be close to what you're looking for