How to play music from my computer on iPhone (without iTunes)?
You want a DNLA streaming app for your iPhone, which can play a music library via a DLNA server running on your PC.

For the PC side, google "windows DLNA server" will get you current choices, but at this time there are Plex, Tversity, and Twonky

On the iPhone side, look at apps like Media:connect. There are summaries of current choices online, like this one: http://blog.dreamcss.com/iphone/dlna-streaming-apps-for-iphone/