How did people get rid of their waste in the Elizabethan era?

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.