How do I read a string char by char in C++?

The simplest way to iterate through a string character by character is a range-for:

bool Class::func(const string& cont){
    for (char c : cont) {
        if (c == '(') { ap++; }
        if (c == ')') { ch++; }
    }
    //...
};

---

The range-for syntax was added in C++11. If, for some reason, you're using an old compiler that doesn't have C++11 support, you can iterate by index perfectly well without any casts or copies:

bool Class::func(const string& cont){
    for (size_t i = 0; i < cont.size(); ++i) {
        if (cont[i] == '(') { ap++; }
        if (cont[i] == ')') { ch++; }
    }
    //...
};

If you just want to count the opening and closing parentheses take a look at this:

bool Class::func(const string& cont) {
    for (const auto c : cont) {
        switch (c) {
            case '(': ++ap; break;
            case ')': ++ch; break;
        }
    }
    // ...
}

const string *p = &cont;
int k = 0;
while (p[k].compare('\0') != 0)

Treats p as if it were an array, as p only points to a single value your code has undefined behaviour when k is non-zero. I assume what you actually wanted to write was:

bool Class::func(const string& cont){
    while (cont[k] != '\0') {
        if (cont[k] == '(') { ap++; };
        if (cont[k] == ') { ch++; };
        k++;
    };
};

A simpler way would be to iterate over std::string using begin() and end() or even more simply just use a range for loop:

bool Class::func(const string& cont){
    for (char ch : cont) {
        if (ch == '(') { ap++; };
        if (ch == ')') { ch++; };
    };
};

If you want to copy your string simply declare a new string:

std::string copy = cont;