Like all hand washing rituals in Jewish tradition, one is supposed to use a wide cup filled with clean water. Unlike the way one washes one's hands for bread or other food items, the morning washing is done three times while alternating hands: right, left, right, left, right, left. Some people have the custom to repeat the washing pattern a fourth time as well. After the hands are washed, they should be thoroughly dried.
Tradition notes several reasons for washing one’s hands immediately upon waking (after reciting Modeh Ani). One of the most commonly cited reasons is to remove the impurity of having come in contact with partial death that lingers after sleep, since sleep is considered to be like 1/60th of death.
There are different opinions regarding how quickly one must wash one’s hands in the morning, but it is generally considered that one should not walk more that 4 amot (6 - 8 feet) without washing. Many, therefore, have a custom of keeping a washing cup and bowl by their beds. There are, however, numerous rabbinic opinions that permit a person to walk to the nearest sink, if it is not too far from one’s bed.
As with all points of Jewish law and ritual, there are basic parameters and different customs that define how the ritual is performed. It is therefore best to discuss the subject with your local rabbi.