How do I check apps running in background.

With a few exceptions, iPhone apps don't run in the background. Notable exceptions would be music apps and navigation apps.

Now, apps can update data in the background. You can see what apps are set to do that by going to Settings>General>Background app refresh.

But the bottom line is that unless your phone is turned off, it's doing something. So, if you have concerns about data usage, turn cellular data off when you're not using it or put the phone in Airplane Mode (that won't turn off WiFi).