Your current setup should be fine with an Ethernet cable from the PC to the router and a Wifi connection from the laptop to the same router.
You can check the connection exists at a low level by opening a Command Prompt window (in Windows search box, enter CMD) and typing in IPCONFIG on each computer.
You should then see the IP Address assigned by the router to each machine.
When I do this on my desktop PC, under "Ethernet adapter Ethernet", it shows 192.168.0.8 as the IP address of my desktop.
On my laptop, typing in IPCONFIG shows the IP address of the laptop is 192.168.0.4.
Now that I know the IP addresses of each machine, I can check they both "see" each other over the network. N.B. My desktop is 192.168.0.8 and my laptop is 192.168.0.4. Your IP addresses will probably be different, e.g. they may start 192.168.1.xxx, or 10.0.0.xxx.
If I type PING 192.168.0.8 into the Command window on the laptop, the laptop "pings" the desktop PC with 4 32-byte message blocks (similar to four ASDIC pulses from a destroyer hunting a submarine).
If the desktop PC receives the "ping" from the laptop via the network, it replies and I get a message saying "Packets sent = 4, Packets received = 4, Packets lost = 0". This confirms I have a solid network connection via WiFi and Ethernet.
I can also "ping" the laptop from the desktop, by typing in PING 192.168.0.4 in the Command window on the desktop. If the laptop responds, I should see a message on the desktop PC saying all 4 packets of data were received.
If you see "Request timed out", you probably typed in a comma "," instead of a period "." as the separator between the four numbers in the IP address. Alternatively, you may have entered the wrong numbers and pinged an address with nothing at the other end.