You can buy this stuff called Stitch-n-tear, or tear away interfacing, at your local fabric store like Joann's or any other place that carries interfacing for sewing. Trace your drawing onto that, pin it or baste it in place on your item just like that tutorial did with printer paper, and then do your embroidery stitches on the design. When you are finished, carefully tear away the interfacing, holding down the embroidery stitches as you go along, and you will be left with the design stitched to your knitted item. this works on all sorts of fabrics, like velvet for Irish step dancing dresses that I showed on my site many years ago: http://www.hookedonneedles.com/2008/...with-hoop.html
For added stability, you could also use a piece of regular interfacing (not tear away) on the back of the knitting which you would leave after the embroidery is finished. You've probably seen this on mass-produced embroidered items. You could even use a hoop as illustrated in that embroidering on velvet link to ensure that you don't pucker or stretch your knit fabric in the embroidery process.
Very cute idea for a kid's hat! Are you going to use a drawing by the child? That would be super cute to let the child draw something, then trace it onto the tear-away interfacing and embroider it onto their hat or jumper or tote bag or pillow. Lots of possibilities!
Post a picture of your finished project so we can all see!