Unless you're rooted, I don't think you can. If you're rooted, all files should appear on the File Explorer when you connect your phone to the PC (from what I understand, not really sure as I'm not rooted myself). All you could backup is your files (pictures, music, documents...). Or if you have a PC suit from the phone's manufacturer, you could connect that to the PC and do a backup of the phone in that software (sort of like iTunes), but even then, I don't think it backs up app data.
In any case, when you switch phones, the moment you activate your Google account in the new unit, it will ask you if you want it to automatically download all your apps. You do lose app data like game saves and such (but really, in this day and age, if a dev doesn't implement Cloud Save support in their games, it's just lazy on their part and therefore I just don't spend my money on their games).