Q: How do I remove apps running the in the background

tap home button twice then hold icon at the bottom of screen until wiggle and then tap the minus signs.  When you have closed them all tap home button once

Double-click on the Home button which will bring up a row of apps currently running. Press on one until the all wiggle. Then tap of the circled "x" in the upper-left corner to kill. Use your finger to scroll the icons.

Very, very few apps actually run in the background.  Only a handful of apps such as audio streaming and navigation are actually doing anything. The vast majority are simply frozen for rapid task switching and consume no battery or runtime memory.

iOS: Understanding multitasking

You might also want to read this http://speirs.org/blog/2012/1/2/misconceptions-about-ios-multitasking.html

Double-press the HOME button. The Quick Launch ribbon will appear at the bottom of the screen. The apps displayed are NOT necessarily running in the background; most of them are suspended. But a few may be. To remove them hold any app until a "-" appears, then tap the "-".

You cannot kill any of the built in apps this way; they always run in background, and if you remove them they restart automatically.