Do you have the latest version of Flash Player? When you have a You Tube video playing right click and then click on "Settings" - if the "enable hardware acceleration" is checked, then uncheck it.
You don't say what OS you are using but to play HD in iPlayer requires
"A modern computer with a graphics card and processor than can handle the graphics, otherwise playback will be choppy"
this is from the iPlayer site. Your broadband speed should be fine as the minimum requirement is 3.2mbps.