Yes.
1)
Download Project64k and extract it somewhere. Where you extract it, make a folder for roms.
2) Download
USA version of Mario Kart 64 and extract it, putting the .n64 file into the rom folder you just made
3)
Start project64k via Project64k.exe (With the K, NOT WITH project64.exe)
4) Go through all the settings etc. First, go on file -> choose rom directory, and set your rom folder. Now mario kart should be picked up and listed in the main window. Then go to options and configure graphics, input to set your keys/pad etc.
5) Choose file -> start netplay... and it will start to list servers. Let it finish, or at least wait until you find a server with under 60ms ping. Also, choose a username and type it into the username box.
6) Find a server that seems good (probably best to look for the lowest ping, and one that is empty for convenience. Use
RMRK's IRC channel to share which server to connect to etc. Join the server of choise with the button. Now you're at the server's chat / lobby thing.
7) Either wait for somebody to start a game, or use the create new game button to select a game of yours and start it. It's self explanatory.
8) Have fun playing! The order that players join/start determines which player they are, i.e. the 3rd player to join would be emulated as the guy with his controller plugged into port 3 on the N64.