The liek comes from a misspelling of like, while the omgz comes from a weird z-addition of omg which stands for oh my god or oh my gosh. Usually used by people who slam their heads on the keyboard and cannot spell for shit, e.g. valleygirls, preppy people who type in sticky caps, or 12-year-olds.
<girl one> liek omgz diz guy spred a rumer abut meh..,,,
<girl two> omgzzz!111 wut a meenee!!11
<girl one> i nooooooo
<girl two> i no u shud haff sbex wiff his gf hed get reeel mad!111 majer paybek
<girl one> wowww liek omgz u r teh so smaert!111 thx babiiiiii