public partial class Form1 : Form
    {
        public Form1()
        {
            InitializeComponent();
            Graphics g = panel1.CreateGraphics();
            Field field = new Field(g);
            field.DrawField();
        }
    }

    class Field
    {
        private char[,] Cells;
        Graphics g;

        public Field(g)
        {
         this.g = g;
             //...
        }

        public void DrawField()
        {
            for (int row = 0; row < 6; row++) {
                for (int col = 0; col < 7; col++) {
                    this.DrawCell(Cells[row, col], row, col);
                }
            }
        }

        private void DrawCell(char cell, int row, int col)
        {
            g.FillRectangle(Brushes.White, x, y, cellWidth, cellHeight);
        //...
        }
    }
}