Converting a string to an Integer in C

Avoid mysterious bit-shifting

return ~output + 1;

Is the same as:

return - (output + 1) + 1;

And as:

return - output;

But the last one is way more obvious than the first one.

Do not hide variable declarations

int ASCIICodes[] = { '0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'A', 'B', 'C', 'D', 'E', 'F' };  _Bool negative = 0;

If I look at the above line, the least thing in my mind is that a variable is defined after that list.

If a user has a small screen or a big font size he will be very puzzled when he will see you use negative without seeing its definition.

Just use a newline, they are free:

int ASCIICodes[] = { '0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'A', 'B', 'C', 'D', 'E', 'F' };
_Bool negative = 0;

Make variables as local as possible

I used x++ to express that I will skip the first character:

if (x[0] == '-')
{
    negative = true;
    x++;
}
else if (x[0] == '+')
{
    x++;
}

This allows me to make count local.

Using for loops when practical

The outer of your loops is a do while, but looking around I found all three components of a for loop, just shuffled into the code.

A for loop will give it more organization:

for (int count = 0; x[count] != 0; count++)
{
    for (int i = 0; i < base; i++)
    {               
        if (ASCIICodes[i] == x[count])
        {
            output = output * base + i;
            break;
        }
    }
}

Simplify the initial conditional

Assigning a boolean to negative directly and using a || conditional looks shorter and simpler:

We go from:

_Bool negative = 0;
int count = 0, output = 0;  
if (x[0] == '-')
{
    negative = 1;
    count = 1;
}
else if (x[0] == '+')
{
    count = 1;
}

to:

bool negative = x[0] == '-';

if (x[0] == '-' || x[0] == '+')
{
    x++;
}

While preserving identical functionality.

it's vs its

... How to choose between it's and its?

This is actually really easy, do you mean "it is" or not?

Use it's when you mean it is.

// Each element is the ASCII for it's index. 

Here you should write its because you mean possession, intended as a property of the element.

Saying that:

// Each element is the ASCII for it is index.

Makes no sense.

So the correct comment will be:

// Each element is the ASCII for its index.