First of all, you have missed : after private
Second, if method (String name) in the cpp file should be the method (String name) from your class, it must be: 
bool me::method(std::string name)
{
    // ...
}
Third, if you want this bool me::method(std::string name) to be different function, a global one, not from you class, it must be:
ret_type method(std::string name)
{
    // ...
}
And, fourth, 
cout<<"name"<<endl;
will pring the string (literal) "name". If you want to print the variable name, use it without the quotes:
std::cout<< name <<endl;
I'd recommend you to get a book
Ah, and this one:
me::me()
{
    method(string name); //can i do this? isn't there another alternative?
}
method(string name) - this is not valid syntax. It should be something like:
me::me()
{
    string name;
    // do something with name
    method( name ); // if "method" is a method, for real
}