Generally speaking, a microcontroller is way underpowered to be able to withstand the slams it would get from a direct connection to the Internet. In case you want to do this, you'd also have to implement TCP/IP on the microcontroller because that is the protocol spoken on the Internet. So, you most probably don't want to connect your microcontroller to the Internet that way.
What I would do: buy a SOHO Wifi AP/router box, which is capable of running Linux, and has a USB host port. Connect that box to the Internet, connect the microcontroller to the box through the USB port, with a TTL-level USB-serial adapter, and run the software interacting with the microcontroller on the box. The TCP/IP part is handled by the OS (Linux), and the web interface is provided by the software running on the box. And the microcontroller would only have to handle a serial connection, which is way easier to program.
Even though you could use ENC28J60 to connect your microcontroller to the box through an Ethernet LAN connection, there's usually no point in doing so: most microcontrollers provide a serial interface, and programming the serial interface on the microcontroller is a lot easier than programming the Ethernet interface.
Examples for the components:
SOHO router: Asus WL-500gp, TP-Link WR1043ND
Linux distribution: OpenWRT
TTL-level USB-serial adapter: Ebay