There are two options to build: either get an old used standup
or cocktail cabinet arcade machine, gut the contents and insert
your own computer, monitor, and controls. Or, build your own
cabinet from scratch. You can find plenty of free plans on the
internet, or you can purchase prebuilt ones.
Now, the cost. The most expensive is buying a pre-built cabinet,
they're well made, but can cost upwards of $2500. A used old
cabinet can be found for many prices in want-ads and craigslist,
you just have to pick up the item yourself. About $100 to $800 for
the really nice ones. Building your own from wood is the cheapest
alternative, and the most work involved. From $50, to $350
depending on what you want.