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The Stream class has pure virtual methods which must be implemented in derived not abstract class.

The pure virtual method from base class Print is:

virtual size_t write(uint8_t) = 0;

The pure virtual methods from Stream are:

virtual int available() = 0;
virtual int read() = 0;
virtual int peek() = 0;

additionally add in your class the line

using Print::write; 

to pull in overloaded methods write(str) and write(buf, size) fromPrint

Here I have an example of a simple class derived from Stream.

The Stream class has pure virtual methods which must be implemented in derived not abstract class.

The pure virtual method from base class Print is:

virtual size_t write(uint8_t) = 0;

The pure virtual methods from Stream are:

virtual int available() = 0;
virtual int read() = 0;
virtual int peek() = 0;

additionally add in your class the line

using Print::write; 

to pull in overloaded methods write(str) and write(buf, size) fromPrint

Here I have an example of a simple class derived from Stream.

EDIT: Most of your question seemed to focus on wrapping some Stream implementation. Now I see that you maybe want to enhance the SoftwareSerial class and add a new method in the inherited class. It is simple:

MySerial.h

#ifndef _MYSERIAL_H_
#define _MYSERIAL_H_

#include <SoftwareSerial.h>

class MySerial: public SoftwareSerial {
private:
  int param;

public:

  MySerial(uint8_t receivePin, uint8_t transmitPin) :
      SoftwareSerial(receivePin, transmitPin, false) {
    param = 0;
  }

  void resend(char c) {
    write(c);
  }
};
#endif

MySerial.ino

#include "MySerial.h"

MySerial mySerial(6, 7);

void setup() {
  mySerial.begin(9600);
}

void loop() {
  if (mySerial.available()) {
    int c = mySerial.read();
    mySerial.resend(c);
  }
}

The Stream class has pure virtual methods which must be implemented in derived not abstract class.

The pure virtual method from base class Print is:

virtual size_t write(uint8_t) = 0;

The pure virtual methods from Stream are:

virtual int available() = 0;
virtual int read() = 0;
virtual int peek() = 0;

additionally add in your class line

using Print::write; 

to pull in overloaded methods write(str) and write(buf, size) fromPrint

Here I have an example of a simple class derived from Stream.

The Stream class has pure virtual methods which must be implemented in derived not abstract class.

The pure virtual method from base class Print is:

virtual size_t write(uint8_t) = 0;

The pure virtual methods from Stream are:

virtual int available() = 0;
virtual int read() = 0;
virtual int peek() = 0;

additionally add in your class the line

using Print::write; 

to pull in overloaded methods write(str) and write(buf, size) fromPrint

Here I have an example of a simple class derived from Stream.

The Stream class has pure virtual methods which must be implemented.

The pure virtual method from base class Print is:

virtual size_t write(uint8_t) = 0;

The pure virtual methods from Stream are:

virtual int available() = 0;
virtual int read() = 0;
virtual int peek() = 0;

additionally add in your class line

using Print::write; 

to pull in overloaded methods write(str) and write(buf, size) fromPrint

Here I have an example of a simple class derived from Stream.

The Stream class has pure virtual methods which must be implemented in derived not abstract class.

The pure virtual method from base class Print is:

virtual size_t write(uint8_t) = 0;

The pure virtual methods from Stream are:

virtual int available() = 0;
virtual int read() = 0;
virtual int peek() = 0;

additionally add in your class line

using Print::write; 

to pull in overloaded methods write(str) and write(buf, size) fromPrint

Here I have an example of a simple class derived from Stream.