The first way you can close Skype is to double-click your home button. The lower part of the screen will open. You should see the Skype icon there. Press and hold it and when it either wiggles or has an "X" in the left corner (or both), you can press the X - Skype will close.
In addition, go to the Settings app and find, in the left column, the Skype app. Then, on the right side, you will see an option to "Go offline." Pressing that line will take you to some options, one of which is "immediately" (Skype will then close when you close the app). Now, with this, it will still be in the multitasking bar (from the 1st paragraph) but it will no longer show you as online. Once you've logged off, if you wish to close it all the way - just do the step in the first paragraph.
Hope this helps.
Marilyn