You are not enabling the timer peripheral's clock, from what I can see.
Note that your code does this:
RCC_APB2PeriphClockCmd(RCC_APB2Periph_GPIOC | RCC_APB1Periph_TIM6, ENABLE);
^ ^ ^
| | |
APB2 APB2 APB1?!!
but this can't be right; you're using constants for peripheral clock 1 and 2 in the same call, to clock 2. That won't fly.
You need to have:
RCC_APB1PeriphClockCmd(RCC_APB1Periph_TIM6, ENABLE);
RCC_APB2PeriphClockCmd(RCC_APB2Periph_GPIOC, ENABLE);
You really should be using the Standard Peripheral Library for the timer initialization too, no point in poking registers directly.