How to detect who is downloading apk files in background?
Probably the quickest fix is going to be when it downloads an .apk, go to the notification, long press it, and when "App Info" pops up, click it. It will open in the application manager the app that downloaded it. (It may not be available on your phone, but it works on my S3.)

Your application manager is another wealth of information. There are way too many specifics to get into much detail about, but for example an app that is supposed to be offline, look at it's data usage. If it has used several MB of data, there's your culprit.

You could also narrow down the suspects by looking at the permissions. If it doesn't require network access, it can't download anything. (Although if this doesn't work, I'm not sure how trustworthy the permissions tags really are if we're dealing with a Trojan).