The Angles, the Saxons and the Jutes, Germanic tribes, all came over to Britain after Rome abandoned the island, which is generally dated to around 410. (These names are given by Bede and don't completely reflect the diversity of Germanic tribes but it's convenient). It is believed that they began the "invasion" of Britain when they were invited by native Britons to fight the Picts and Scots and the rest invading from the North and Ireland. However, the Saxons had been raiding Britain for quite some time. The Romans had set up fortifications to protect against Saxon invasions (called the Saxon shore) between roughly 270-290. But after they were gone there was no stopping them. There are chronicles that say the land was under Saxon rule by 441. Exact date is questionable but it gives you an idea of the time frame.

For a long time it was thought that these tribes all sort of invaded Britain in a rush and mostly wiped out the native population. But the consensus on that has shifted a lot. It is now believed that they came over in waves. And there was certainly quite a bit of fighting back and forth. And archaeology supports that there were periods when the British gained the upper hand and pushed these Germanic tribes back a bit. In the long run, the Saxons, Angles, and Jutes won. They set up several kingdoms through out Britain. Wessex (West Saxony), Essex (East Saxony), Sussex (South Saxony), Mercia, Kent etc. They weren't one monolithic group of people. And that's why they set up several different kingdoms (and the kingdoms were often at war) rather than a nation. Although they certainly shared a culture and a language.

The legend of King Arthur actually dates back to this time period. He was alleged to be a native Briton who was able to fight the Saxons et. al. and won repeated battles. But his wins were reversed with his death. Whether there is any truth to that is beside the point. It gives you an idea of the landscape that such a legend would appeal to the native people (much, much later the Anglo-Saxons embrace Arthur as their own).

But there was also a lot of inter-marriage between the Germans and the native Britons and some of the British culture and language was retained. So, scholars now think it wasn't necessarily the complete near extermination that people used to think. Also a lot of Britons fled to France (Brittany) and Wales. The native British culture (with adaptions and changes) was alive there long after it had mostly died out in England, proper. The English language as we know it is an amalgamation of a lot of languages but at its root it is based on these Germanic languages brought over by the Saxons and co. Not based on the language of the Britons.

By the time of the Viking raids, these English kingdoms were established and settled in. They were no longer fighting for any land from the natives and hadn't for centuries. They absolutely saw themselves as the native and rightful people of England (although they wouldn't use that name). But the concept of a united England didn't exist. So they saw them selves as Saxons or Mercians and so on.

When the Vikings came it was a crisis. The first major raid was in 793. At first, it was just raids but eventually some of the Norse began to settle down. And then there became a general Norse push to establish kingdoms there of their own. They also did this in Ireland, by the way. By the time of Alfred they had conquered almost all of the English kingdoms except Wessex and that's when Alfred the Great comes in. And (spoiler!) he is able to conquer most of England from the Vikings. But the Vikings retain two kingdoms to the North, East Anglia and Northubria, which becomes the Danelaw (the idea of a jury comes from this, by the way). His successors later conquered these areas but the Danes remained without causing anyone much trouble until many were slaughtered by Aelthred the Unready. That was Sven Forkbeard's "justification" for conquering England.

As to the relation to the Vikings, the Saxons and co. were Germanic tribes. And the Norse/Vikings are also Germanic tribes of a sort, in the sense that they had a somewhat similar language and had once shared the same gods (but by this time England was Christian and there was no common religion). There had been some exchange and cross-culture in the past (the Anglo-Saxon epic of Beowulf is mostly centered in Denmark) but that was long over. They weren't related in any significant way by the time of Alfred. Neither group felt any kinship toward the other.

As a sort of cap to all of this, well after Alfred the Norse finally DO win and take over England when the Danish king Sven Forkbeard conquers it. And his son Cnut rules for a long period of time. And then after a brief return to Anglo-Saxon rule under Edward the Confessor the Normans show up and take it all over. The Normans were viking descendants. So, in the end, long after the period the show will cover (so not a spoiler!), the vikings sort of do win.

Moprollems has explained it well, but I'd like to add a little, to put it in the context of that show and wider history. Several hundred years before the Vikings invaded and conquered the north (Danelaw), the Angles, Saxons and Jutes had come from areas around what is now Germany, and culturally taken over what is now known as England. Historians have argued about how this take-over worked; was it a conquest (the traditional view) or more a process of cultural assimilation (the revisionist view). Most historians now think it was more the second. There was a lot of intermarriage (genetically, English people are descended from both Britons and Germanics), but the Anglo-Saxon culture and language became dominant. If you study German, you'll find that it's a reasonably similar language to English, especially in grammar and basic words ("My name is Bob", "Mein Name ist Bob").

The Kingdom of Wessex, that is featured in the show, is etymologically descended from "West Saxons", while east of it is an area (where I'm from) called Essex; "East Saxons".

These Germanic tribes were pagans (whose religion was somewhat similar to that of the Vikings: we get our English day-names from their Gods, such as Tywaz's Day, Wodan's Day and Thunor's Day). I think in the show the protagonist mentions that his ancestors were pagans and worshipped Thor; he definitely does in the book (though what they would have worshipped is Thunor, the west Germanic version).

The reason Wales is not England, is that it was part of Britain that was not taken over by Anglo-Saxons, and so kept its Briton culture, language, and Christian religion. In fact, in ancient Saxon, "Welsh" means "stranger" or "foreigner". The Welsh continued to refer to themselves as British (Welsh being an English word) though later they changed to a different word (Cymraeg).

Another part of Britain that was not taken over by the Anglo-Saxons was Cornwall. This is why in the show, they raid Cornwall, which was not considered English. The people spoke a Brythonic language (Cornish), and were Celtic, not Germanic (in fact, some people in Cornwall still speak Cornish).

Later (before the Vikings came) the Anglo-Saxons converted to Christianity, and began developing a pan-English culture (as opposed to considering themselves Angles or Saxons).

This process was heightened by Alfred the Great's conquest of all of England (sorry if that spoils the next season for you!) as for the first time, the whole country (not including Cornwall, which was not considered England) was united.

Tldr; Britons got invaded by Romans. Romans went home. East Britons got conquered/assimilated by Anglo-Saxons. This area became known as England, while the other parts became known as Wales and Cornwall. Vikings conquered north England, eventually destroying all English kingdoms but Wessex. Alfred beat the Vikings and united England for the first time under one King.

The Anglo-Saxons weren't one group so much as a variety of Germanic groups coming from lands (such as Saxony) that later colonized what we now know as England. The English today aren't Celtic nor primarily related to the "Britons" that inhabited England earlier (though the Saxons did interbreed with native celts, it was Germanic culture that won out)

Even the name "England" is essentially the result of all the changes to the language that would later become English and may have it's root in the words used to described the peoples' there (Angelcyn, though after centuries of rule by scandinavians and some french with latin thrown in it's not hard to see how it could mutate into "English")

As for how they got there, one of the more popular theories is that they were invited by local Celtic kings as mercenaries to deal with rival tribes to the north and given a plot of land in what is now known as East-Anglia. Over time and some local conflicts they came to rule what is now England and grew to be the dominant cultural group there.

While they have many similarities to Vikings, they weren't the same group of people who were raiding Catholic Monasteries and are who we think of when we think of "Vikings".

EDIT: It's important to note that the Anglo-Saxons later did battle with the vikings who'd conquered much of England, and it was thanks in part to what was essentially a campaign of vicious guerilla warfare by King Alfred The Great that England was united under one rule. He threw the vikings out of his country and it would continue to be ruled by Anglo-Saxons up until the Norman Conquest.