Q: How do I clear apps running in background?

iOS 7 How to Close Apps

Double Tap the Home Button... Then swipe the App (not the icon) Upwards... Tap the Home Button when finished.

More Info on Multitasking Here  >  http://support.apple.com/kb/HT4211

Most are not actually running in the background but to close them, double tap the home button, swipe up on the screen shot, tap the home button when done.

Not much need to do that, but:

Most apps do not run in the background, but this is how you can close them.

for iOS 5 or 6...

Tap the home button once. Then tap the home button twice and the recents tray will appear at the bottom of the screen. Tap and hold down on any app icon until it begins to wiggle. Tap the minus sign in the upper left corner of the app that you want to close. Tap the home button twice.

for iOS 7...

To close an app, drag the app up from the multitasking display. Double tap the home button and you will see apps lined up going left to right across the screen. Swipe to get to the app that you want to close and then swipe "up" on the app preview thumbnail to close it.