[Valmont]

This is better and should be done:

The header:

//Implicit conversion and default constructor.
MyClass(const char* = "");

The implementation:

#include <cstring>

MyClass::MyClass( const char* s )
   : m_pChars( new char[ strlen( s ) + 1 ] )
{
  std::strcpy( m_pChars, s );
}

Don't forget the destructor!

The wrong code:

MyClass::MyClass() : 
myString[0]='\0';
{}

You may NOT use "=" here. You must use the parenthesis to explicitly denote the initialization in a constructor. Outside the (copy)constructors, you may use "=" to denote initialization indeed.

There is a second major topic here:
Your code (and Redhead's) implies that myString has a fixed size set in as a data member in the header. But this is illegal for arrays UNLESS it is a static member. LEGAL code below:

#include <cstring>
#include <iostream>

class MyClass
{
public:
  static char myString[5];
};

char  MyClass::myString[5] = "test"; //OR "" to initialize it as empty.

int main()
{
  std::cout<<MyClass::myString<<std::endl;
  std::cin.get();
  return 0;
}

Conclusion
Use my method (the very first example).