Can I transfer music to my iPhone 5s?

So I want to connect my phone to my laptop which isn't anything to do with Apple, in order to transfer music for free however I want the music to be put into my music 'app' as in the one where my bought songs go in to, is this possible? If it is how do I do it?

Yes, download itunes on your laptop. You can sync music downloaded from anywhere, not just bought on iTunes. Drag the music files to iTunes to add them to your library, and then choose to sync them to your iPhone.

Once you connect the iPhone, go to the music tab in the iPhone section and choose what you want to sync, after that, hit apply and it should sync the music over and it'll be in your music app.

You can even choose to let iTunes order your music in your computer going to preferences, this way you won't have the files all around your laptop but in one folder in Music/iTunes.