One mouse and keyboard for a Linux and Windows computer
The solution could be a switch, a Keyboard-Video-Mouse switch, or a KVM-switch. Actually I just need a keyboard and mouse for two computers. No video-switch needed, because I use only the screen of one computer, and connect the second computer with a HDMI cable to this screen. In this way I can switch between the sources of the screen to display one computer at a time. I choose to buy a license of the synergy-tool. 
This tool, installed on both computers, communicates the mouse and keyboard-events through your local network. So one computer has to be the server and the other has to be the client.
In my setting, with a VPN on the server-computer, this didn't work, because the VPN apparently takes all the traffic out of the local network, and to get back to a local address will not be possible. Some VPN's have the option to share on the local network. You have to opt for this to let Synergy do its amazing work.