Here is a char wrapper class called Char. The two examples in main() demonstrate that you can use Char-typed values just like you would char values, except that Char has a [] operator, for getting its value's bit at some given index.
#include <iostream>
class Char {
    char c;
public:
    Char() = default;
    Char(const Char&) = default;
    Char(char src) : c(src) {}
    Char& operator = (char src) { c = src; return *this; }
    operator const char& () const { return c; }
    operator char& () { return c; }
    // Special [] operator
    // This is read-only -- making a writable (non-const)
    // version is possible, but more complicated.
    template <typename I>
    bool operator [](I bit_idx) const { return !!(c & (char(1) << bit_idx)); }
};
int main() {
    // Example 1
    // Initialize a new Char value, just like using char.
    Char my_char = 'x';
    // Math operators work as expected
    ++my_char;
    // And cout will produce the same output as a char value
    std::cout << "Bit 3 of '" << my_char << "' is ";
    // But unlike a char, the [] operator gives you
    // the bit at an index, as a bool value.
    std::cout << my_char[3] << "\n\n"; 
    //Example 2
    // Specify the Char type in a range-based for loop to
    // iterate through an array of char values, as Char values.
    const char str[] = "Tasty";
    for(Char ch : str) {
        // check if value is nonzero, the same as you would a char value
        if(ch) {
            // Send the value to cout,
            // cast to an int to see the ASCII code
            std::cout << ch << " (" << static_cast<int>(ch) << ") ";
            // Count down from bit 7 to 0 and use
            // the special [] operator to get each
            // bit's value.  Use this to output each
            // value's binary digits.
            for(int bit=7; bit>=0; --bit) {
                std::cout << ch[bit]; 
            }
            std::cout << '\n';
        }
    }
}
Output:
Bit 3 of 'y' is 1
T (84) 01010100
a (97) 01100001
s (115) 01110011
t (116) 01110100
y (121) 01111001