As far as I have read, a couple of Separatists (John Greenwood and Henry Barrowe) were executed in 1593 (near the end of Elizabeth I's reign) for advocating Separatism. After an intense crackdown, their followers fled to the Netherlands which eventually formed the basis of the Pilgrims of the Plymouth Colony.

Other than that, Separatists were numerically insignificant throughout Elizabeth I's reign because the bishops were pretty successful in convincing the Puritans that the Church of England was not unreformed. Still, there were those who seeked more reforms and they evolved their own forms of worship and disciplines though not intending to be divisive. Only towards the 1630s did the Puritans believe that the bishops had moved towards a different path that Separatism really did grow and became a bigger problem.

Elizabeth I and Religion 1558-1603 By Susan Doran

Access to History: Elizabeth I Meeting the Challenge: England 1541-1603 By John Warren

She issued an edict against Nonconformists in 1573 (a copy is here: http://www.thereformation.info/Nonconf1573.htm) but she never really tended to actively persecute groups who weren't overtly threatening. To be honest, nonconformity (Puritanism) wasn't really underway during her period. She did take some actions against Puritans who sought to impose their views on the wider church established under the Elizabethan Settlement (essentially a Protestant church but with certain Catholic elements intact which the Puritans detested). Some Puritans relocated to the Netherlands during her reign but never in large numbers. Mostly, society thought of Puritans as rather odd extremists from my perspective, who were resented for seeking to impose their morality upon the rest of society. Frankly, I am sure their departure to the distant new world was greeted with a sigh of relief from those sick of being lambasted and preached at.

For a little background, England had been going through a great period of religious turmoil since the establishment of the Church of England under Henry VIII, most recently under her half-sister Queen Mary, who had been a Catholic and had viciously persecuted religious dissenters. Under Elizabeth, the Church of England was reestablished as an independent body with Elizabeth at its head. Now, because she didn't want to be seen as another Mary, there was no criminalization of Catholic beliefs.

That being said, all religious and secular officials had to swear loyalty to her as both queen and head of the Church, meaning that anyone who refused to do so out of loyalty to the See of Rome lost their jobs. Furthermore, laws were enacted to the effect that everyone was required to attend church, and all congregations were required to use the Book of Common Prayer prescribed by the Church of England. This was a serious injunction against any who held heretical beliefs, and it's important to note that heresy itself remained a crime for which people were often imprisoned or executed, the latter punishment being more common in cases where they were distributing literature or attempting to convert others.

English Separatists (itself an umbrella term for various different religious groups) were not necessarily theologically in opposition to the established Church, more tending to disagree with its practices. A major part of the issue was that many wanted a Congregationalist model, in which each church was independent of any other,as opposed to the hierarchical model inherited from the Catholics. In this case, Separatism refers to the desire to be separate from the established church, which was a threat to its civil authority at a time when the distinction between the secular government and the church was quite blurry. The group we today might call the Pilgrims left England in order to be able to form their own church independent of this authority.

To answer your questions; it's hard to say exactly how she viewed them personally, but in her office as Supreme Governor of the Church of England, she was surely not a fan. They were not allowed to have separate congregations. In terms of toleration, they were in fact tolerated so long as they didn't speak up, share their beliefs publicly or threaten the established church, which doesn't sound that great until you compare it to the reign of Mary, when religious dissenters were hunted down and burned at the stake.