How can I know when my computer is pinged?

Your aftermarket Windows firewall should have an option to Log any kind of network activity. Find ICMP (the parent protocol of Ping) and set it to Log instead of just Deny or Accept. A really good firewall will allow you to drill down into ICMP and just log Pings, since there are other types of ICMP packets.

There are different ways to do it, but the easiest would be to install Wireshark on the computer so you can do a packet capture, and put a filter on the capture for icmp traffic only. The capture filter would simply be icmp, can be set in capture options.

Other options would include setting up a mirror port on your switch, and maybe debugging on the router if it goes through a router. But with these, you wouldn't be 100% sure the packet got to the computer (Although if you see the replies, seems to be a safe assumption).