I think you didn't read enough of the data sheet.
Start on page 50 of the data sheet. Now on page 51, you will see a section entitled "Configuring the Pin". Then read Table 20 Port Pin Configurations.
Your code is correctly setting PC0 to output BUT as a sink. Your schematic does not match this. If you are using a sink your LED should be connected appropriately to Vcc and not to GND.
Or you can change your code to something like:
(Verify the names, I didn't compile check them.)
/* set only one bit, set to 1 direction output, read Page 51 2nd paragraph */
DDRC = (1 << PC0);
/* set only one bit, set to 1 to drive it high, read Page 51 3rd & 4th paragraph, refer to page 52 Table 20 */
PORTC = (1 << PC0);
/* embedded apps do not return */
So you have two options change your wiring (and use your original code) or change you code (with no changes to your wiring).