I am wondering if I need to purchase an accounting program to keep track of my first property? It is just one more expense, but since it is just one multi-plex is it really necessary? Also, do I need to file a DBA with the county clerk?

A well organized spread sheet is all you need, but make sure you use it. Do not think that you will remember what that receipt is for, you won't. 

As far as dba, llc or any other, check with your accountant/lawyer. They will let you know. A lot depends on how you purchased the property, cash or financed, a good accountant can advise.

A spreadsheet is probably fine for one property, unless you are planning to scale rapidly, in which case I would recommend investing in QB right away.  

Its better to develop good bookkeeping habits when the portfolio is smaller, so you don't have to deal with a giant mess down the line.