So now that I've figured out how to get what I want, I'm just hoping somebody can let me know a cleaner, less ridiculous way of achieving the same thing. I'm just learning C. Here was my approach.
int main()
{
    // String of positive and negative integer values 
    // Numbers are never more than 2 digits
    char TEMPS[256] = "1 -22 -8 14 5";
    int N = 5;
    int ints[N];
    int i = 0;
    int mult;
    // Arbitrary number to identify that num is not yet in use
    int num = 999;  
    int c = 0;
    mult = (TEMPS[c] != 45) ? -1 : 1;
    while(strcmp(&TEMPS[c], "\0") != 0)
    {  
       if(TEMPS[c] == 32)
       {            
            ints[i] = mult * num;                        
            i++;
            num = 999;
            mult = (TEMPS[c + 1] == 45) ? -1 : 1;
       } 
       else if((TEMPS[c] != 45) && (TEMPS[c] != 32))
       {
            if(num == 999)
            {
                num = TEMPS[c] - '0';
            }
            else
            {
                num = num * 10 + (TEMPS[c] - '0');
            }
       }       
       c++;
    }
    ints[i] = mult * num;    
}
 
    