Hello Lars,
You have an active AVG Internet Security license for 1 computer, valid until 6/1/2018. If your old computer doesn't work anymore, you can install AVG Internet Security on your new device. If not you have to uninstall AVG from the old device and then start the installation on the new PC. It is only necessary to ensure that your license will not be used on more computers than it allows.
To install AVG on the new computer, follow the steps in the article http://support.avg.com/SupportArticleView?urlname=Download-and-install-AVG-Protection
If you face any difficulties with the installation/activation, please provide the screenshot [http://support.avg.com/SupportArticleView?urlname=How-to-create-a-screenshot] of the error message or AVG Interface to check and assist further.