I have two local machines connected to each other via wired ethernet and one of those machines is also connected to a wifi network which provides internet access.

A pseudo-graphical representation of the topology is as follows:

(PC2)----------(PC1)---------(Wifi Gateway)
    192.168.2.x      10.0.0.x

The configuration on PC2 is:

iface eth0 inet static
address 192.168.2.2
network 192.168.2.0
netmask 255.255.255.0
gateway 192.168.2.1

...and the configuration on PC1 is:

iface eth0 inet static
address 192.168.2.1
network 192.168.2.0
netmask 255.255.255.0
gateway 192.168.2.1

On PC1, I've added a default route for wlan0 as I couldn't access the internet otherwise:

route add default gw 10.0.0.1 wlan0

And also tried setting the gateway for the 192.168.2.x network using:

route add -net 192.168.2.0 netmask 255.255.255.0 gw 10.0.0.1

But I still can't access internet from PC2.

Edit

I don't have access to the wifi gateway.