how do i transfer music from iTunes on my computer to my iPhone 4S?

Sync it:

http://support.apple.com/kb/HT1386

And here:

http://support.apple.com/kb/PH2558

You should really read the manual

Open itunes, connect iphone, select iphone, select what you want, sync.

It seems that Syncios iPhone Transfer can help you transfer music from iTunes library back to your iPhone easily. You can transfer music from iTunes to your PC first and then you can transfer from PC to iPhone.

Launch iTunes on your computer and plug into your iPhone to the PC. Click the "File" button and choose to add files or folder from your PC to iTunes library. Click the "Music" icon to sync the audio files. Check the "Sync Music" icon to decide to sync "Entire music library" or "Selected playlists, artists, albums, and genres". If you want to copy the entire music library, check that option and it is done by clicking "Apply". While you want to transmit only certain files, just choose the corresponding option and check the specific groups in the classifications below. After clicking "Apply", the transfer begins.If you want to view the transferred files, click the "Music" button under "On My Device" in the left column so that you can transfer music from iTunes to iPhone 4s.