You can achieve what you want to do, however not with your current wiring diagram. You want PC3 to talk to PC2 while not having access to the PC2 network (which includes greater internet access), which simply will not work as this is.
So, we need to clearly define your two networks and for each PC-to-PC connection you want to have, connect each PC to that network. Assuming I understand your requirements correctly, it should look like this:
Now, the first thing you might say is "Only my PC1 has two LAN/Ethernet ports, not PC2!". There are some solutions for this, but we do need to create some sort of logical-to-the-OS separation of LAN connections on PC2:
If your switch hardware supports VLANs, you could go that route while creating a secondary virtual Ethernet port on PC2 that uses that VLAN. This is the most complicated solution to setup, but requires no additional hardware if yours supports this feature.
Buy a $15 USB-to-Ethernet adapter for PC2. Simplest solution. I use the Plugable brand ones at work.