Hi @franziska_maier
If there are no problems with the screen if you connect the charger first to the laptop and then turn the laptop on and use it, perhaps the battery is failing and needs to be replaced.
Here's an image taken from the [http://s472165864.onlinehome.fr/anyware/manuels/N650DU_ESM.pdf|service manual] for the laptop that shows how to remove the battery. Also in the image I've included the spare part number(s) for the battery. It seems that there are two batteries available for the model. They have the same specifications so they should be both the same. Check if there is a part number on the battery in your laptop to verify if you wish to get the same one.
[image|2768564]
(click on image to enlarge)
Search online using the ''part number only'' to find suppliers that suit you best.