It depends on which wastes you're talking about. If you're talking about sewage, in London they simply dumped the sewage in the streets. The same was true for basic garbage. That meant the city was full of filth, reeked, and was a hotbed of disease. If people lived near the Thames, they'd just dump things in the river.
Now, Hamlet refers to compost, so we can assume that in the country, people composted organic wastes to enrich the soil.
As far as the ultimate human waste (bodies), those were buried.
Methods of human waste disposal in the Elizabethan age varied with people's circumstances. Many houses had latrines that emptied into deep cesspits. These cesspits themselves were usually emptied into middens, the early equivalent of landfills, or nearby streams or rivers. At night, people had lidded chamber pots in their rooms to use for convenience; these were emptied into latrines. Emptying a chamber pot from an urban window onto the street would have been illegal and led to fines. It is portrayed in comedy and satire but does not reflect normal practice.
In rural areas, people may have had outhouses or simply have used chamber pots that they emptied into streams. While working in the fields, of course, human waste would eliminated anywhere convenient. At times it might even be used as fertilizer in the fields.
Most of the time, tenants would throw buckets of waste off of their balconies onto the streets/gutters. This open sewage is what led to so many of the early plagues and deaths in England, but is also reputedly one of the reasons it's chivalrous for the man to walk on the street side of the sidewalk. Since the lady would be under most of the balconies, she would be relatively unharmed by the waste being thrown over. Nowadays, it is assumed that men walk closest to the street to protect women from cars, when actually it was used to protect us from poop.