Contracting vampirisim in Skyrim is similar to the proccess in Oblivion. First, you must come across and fight a hostile vampire (note that not all vampires are naturally hostile). Once you have entered combat, there is a chance to contract the disease with each strike the vampire lands. If you wish to become a vampire, it is advisable to take as many hits as possible. Once you have contracted the disease, the vampire you are fighting can be dealt with.
Upon contraction, you will recieve a text notice in the top left of the screen. This can also be found listed in Active Effects of the Magic panel. You have a period of three days to cure yourself, or the vampirism will set in completely and begin it's stages. You can cure yourself of Sanguinare Vampiris with a Cure Disease potion or by worshiping Talos (shrine in Whiterun).