I modified your code as follows so that it ouputs the board to a user specified file as well.
Code:
#include<iostream>
#include <fstream>
#include<string>

using namespace std;

int main()
{
	string character, b=" ",stop_go;//initialisation of variables
	int size;
	int n=1;

	string fname;
	cout<< "File name please: ";
	cin >> fname;
	ofstream outf(fname.c_str());

	while (n>0)    //while loop so the user can draw multiple times

	{

		cout<<"Give Chessboard Size:"<<endl; //read-in chessboard information and save in variables
		cin>>size;
		//so only odd numbers will be used
		if(size%2==0)

		{
			cout<<"Only enter odd numbers; please try again"<<endl;
			continue;
		}
		cout<<"Give a Character for your chessboard:"<<endl;
		cin>>character;


		
		//begin drawing chessboard
		for(int k=1; k<=size;)   //outer loop for vertical drawing


		{

			if (k%2==0)

				for (int i=1; i<=size;i++)   //inner loop voor horizontal drawing

				{

					if (i%2==0) { cout <<character; outf << character; } 
					else { cout<< b; outf << b;}

				}
			else


				for (int i=1; i<=size;i++)

				{

					if (i%2==0) { cout <<b; outf << b;}
					else {cout<< character; outf << character; }

				}

				cout<<endl;
				outf << endl;
				k++;


		}

		//give user the choice to exit or draw again
		cout<<"Press 'e' to exit or 'd' to try again \n please draw chessboard"<<endl;
		cin>>stop_go;
		if (stop_go=="e") n=0;
		if (stop_go=="d") n=1;



	}


	system("pause");
	return 0  ;

}