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 &lt; 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 &quot;it is&quot; 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.