Is there any way I can use my Wii U to play movies I have on my computer? So if I have purchased a movie through iTunes for example?
You can setup a Plex server on your home PC. Plex server essentially hosts your videos and other media for use from the home, and supports many platforms such as the iPhone/Android, Chromecast, Smart TVs, other PCs, and major consoles.
There isn't a Wii-U app directly to support it, however you can use the Wii-U's built in web-browser with ease.
The basics for setting up a server is beyond a hand-written tutorial for this website, especially as the software changes over time. A quick Google search will need you plenty of high-quality tutorials for your device in question.