...
private TileGame game;

public Form1()
{
    InitializeComponets();

    game = new TileGame(pictureBox1);
}
...

public class TileGame
{
    private Control canvas;

    public TileGame(Control canvas)
    {
        this.canvas = canvas;

        canvas.Paint += canvas_Paint;
    }

    private void canvas_Paint(PaintEventArgs e)
    {
        ...
    }
}