How can I connect a Wii U to my router with an Ethernet cable?

You will need to purchase the Wii LAN Adapter for that. The official adapter is compatible with Wii, Wii U, and Switch. Note that this will occupy a USB port on your console.

According to Nintendo Support, regular USB to Ethernet adapters might not work with the Wii U. This likely also applies to the Wii and possibly also the Switch. You could try using an adapter you already own, but if you're going to buy one for the Wii U, I recommend buying the right one.

You will need a USB to RJ45 adapter. Nintendo explains it here: https://en-americas-support.nintendo.com/app/answers/detail/a_id/1668/~/how-to-connect-a-wii-lan-adapter-to-the-wii-u

There seem to be specific Wii U/Wii USB to RJ45 adapters, but I am not sure if just any USB to RJ45 would work.