But how do I setup the router? Do I use a range extender that can act as a router?
Neither – this has absolutely nothing to do with routing of IP packets.
Conceptually, most home "wireless routers" are 3-in-1 devices: an IP router, an Ethernet (wired LAN) switch, and a Wi-Fi access point. To add more LAN ports, you'd simply connect a second external Ethernet switch to the same router; to expand the Wi-Fi range, you would connect more Wi-Fi APs.
In other words, a you need is a range extender that can act as a range extender (i.e. access point), which is all of them.
As far as configuration goes, only the extender/AP itself needs to be configured:
If you set it to the same SSID and WPA password (but ideally a different channel), then devices will automatically roam to the 'best' AP – which is the most convenient option.
OTOH, if you have mysterious connection problems, then use different SSIDs. (For example, my home printer gets completely confused when it sees the same SSID from two APs...)
(Note: Many people use a 2nd "wireless router" for this purpose, but this usually involves disabling the "routing" part in it. Search for older posts on this topic.)
Do I just plug the external antenna to one of the existing antennas connections?
If the existing router/AP comes with 2 or more antennas, then I would guess that it's optimized for their specific size & distance, and running an external antenna cable off their ports might be a bad idea. (If it has only a single antenna, or a dedicated external antenna port, then that has a bigger chance of working, but you should still read the comments on that linked post.)
Though there certainly are access points built specifically with that kind of usage in mind – usually sold as "for long-range PtP/PtMP connections". Some models accept an external connection, while others have an internal surface of patch antennas. (In pairs they have a range of many kilometers; when used with plain phones/tablets/laptops they're still generally sensitive enough for at least 250–500 m.)
Do I use a RaspberryPi and do something with that?
It's possible to configure any Linux system as an access point by installing hostapd, as long as the hardware supports it. The "Raspberry Pi 3" can do it, but USB-connected Wi-Fi adapters are very hit-or-miss. Running iw phy would show whether AP mode is supported.
(Note that an AP / range extender needs only hostapd bridged to the Ethernet port. Some tutorials tell you to set up dhcpd and/or iptables-masquerade, but you don't need that since your existing router already does this.)
