Some time ago, I wrote a post that touches heavily on this. Check that out for the details, but I'll give you the really basic rehash here.

Royals have excellent paper trails for tracing their ancestry, which allows us to make pretty solid family trees going back hundreds and hundreds of years (keep in mind, of course, that doing so assumes that declared paternity is real paternity, which, over dozens of generations, starts to go against the odds...). In the case of Queen Elizabeth, we can pretty easily trace her heritage back well over 1000 years. The documentation that links her to Charlemagne, who ruled in the 8th and 9th centuries, is pretty well accepted, as well as the links that would trace back to William I of England (here is a family tree that goes back to the 800s). None of that is controversial for the most part (again, discounting cuckoldry). We can go even further than that, though, in which case you start to get into the weird stuff. Norse genealogies, which eventually find their way into the Norman families, as well as Kings of Wessex, start to dip into the Legendary and Semi-Legendary. If taken at face value, QEII is descended not just from mythical persons such as Ragnar Lodbruk (or FX fame), but also the god Odin... If you want to stay within more legitimate bounds, attempts to trace ancestry of Charlemagne (who QEII can in turn trace to) back to Roman times is stuck somewhere between the 4th and 6th centuries, with a few possible candidates. So basically, we are very certain about certain ancestries going at least to the 8th century, but when you go back beyond that, stuff gets fuzzy.

But as I note in the other post, keep in mind that this is also the case for you, assuming you have European ancestry. It all comes together well before we reach that point, so these are your ancestors too. the only difference is that the specific line of QEII remained notable the whole time, leaving us records to connect the dots with.

Elizabeth I or II?

Burke's peerage recognizes Elizabeth II as a descendant of the prophet Mohamed (b.570ish) through Iberian ancestry (however, it is not without serious doubt). Additionally there is good record of Mohamed's ancestry to at least the fourth century. Traditionally he is also thought to be a descendent of Adnan, who died during Nebuchadnezzar II's reign (605 -562 bc). So this is probably the earliest recognized, but not necessarily likely, ancestor.