The clipboard function on the Note series is a support app for Samsung Apps such as S Search, S Voice, Message, etc. To access the clipboard, switch to any of these apps then open the keyboard. Tap and hold the gear/keyboard/mic icon just left of the spacebar and a clipboard option will appear.
You could use message as a clipboard, utilize pen window to float message and manage your clipboard or use a 3rd party app. The GALAXY Note 4 supposedly fixes this problem and the Note 3 should get a retro update from Samsung in the coming months.