Arduino Ethernet Shield: May I connect it to a PC connected to the internet via wifi?
Assuming an Ethernet port on the PC, the Ethernet shield can be connected to it using an Ethernet crossover cable:


(source: Wikipedia)

Many recent PC network interfaces have automatic cross-over (Auto MDI-X), with which even a regular (straight) Ethernet patch cable will work. 

This merely addresses the physical Ethernet connectivity. 

For actually routing IP packets between the Arduino and the rest of the Internet, some software router or software bridge is required to run on the PC. Several open source routing products exist, especially for Linux and FreeBSD, but that is outside the scope of this StackExchange.

Finally, the Ethernet shield does not need a router, it an as well be hooked up to an Ethernet switch or hub.