If the videos are starting to play and then failing to playback properly then it's not likely to be the codecs or software installed that's the problem (although this is possible in theory, see below). It really does sound like your connection. Your computer is easily fast and modern enough to play streaming videos without errors.
Simply, if you moved the computer from school where it was working, to home where it isn't, without changing any settings, then the logical problem is with the home internet connection, based on the symptoms.
Go to www.speedtest.net and test the speed. It should be consistently over 1Mbps and preferably multiple times that speed for streaming video and good general internet. If you find it's lower, or seems fast enough to the speedtest site, but performance is still poor on youtube, then go to start menu, run, type "cmd" then in the black box type "tracert youtube.com" (or the hulu web address instead of youtube.com). if any of the hops show an asterisk (*) or a time higher than the low 100s, then you have a problem connecting to those sites.
Check your modem/router is working correctly if you have any speed or tracert problems as detailed above. Ask your ISP or router manufacturer to help you with this.
Also, just in case there's a software issue, since the videos are flash-based, try downloading and installing the latest version of flash player for Internet Explorer and Firefox:
If you don't have Firefox, try it also as it's good.
http://www.mozilla-europe.org/en/firefox/
http://get.adobe.com/flashplayer/
http://get.adobe.com/flashplayer/otherversions/