Use as high a speed as possible, drill press, carbide bit, gentle feed rate. Don't have a drill press? Get one. A must one of those drill press mounts for a dremel style rotary tool would be great for PCBs, with the high spindle speed.

The circuit is simple enough that you can simply draw the traces onto a piece of board material with a permanent marker exactly as in the schematic, then etch it yourself. No fancy layouts are necessary. Just orient your components as in the diagram & drill the holes in the board material. Then use the marker to draw in the traces. Make sure the marks are thick and black.