While the Anglo-Saxon Chronicle relates quite a series of Viking raids starting from the late 700s (starting with Lindisfarne in 793), the main problem with all written accounts is that they appear to have little archaeological imprint - there might be evidence of the Great Army at Repton in 874AD but that's pretty much it. If you want Viking evidence, it's going to be found elsewhere.
The Danelaw area is the second best place to start looking. After Alfred forced Guthrum into a treaty that separated all of England into two parts, the part administered by the Danes should show us something, but even here the evidence isn't exactly clear. We don't know how many Vikings settled (probably an unanswerable question), so placenames and vocabulary are often touted as evidence - and they probably are. The problem with language-based evidence is simply that - it's the extent of the language, not necessarily the extent of the Vikings or their settlements. The other problem is that the place-names are recorded in the Domesday Book (1086AD) which means you've had 200 years of hybrid living of English and Vikings- so you cannot rule out the fact that a small number of Vikings may have had a huge effect on the language which was taken up by the English.
Viking burials are not excessively common (or rather, they're not very identifiable in the archaeological record) as it appears that most Viking settlers integrated their dress and cultural norms into the English very quickly. Perhaps 25 Viking burials have been found, which is tiny (compare that to the Anglo-Saxon or Roman burials) which probably means they adopted the English forms of burial. Hogback burials might be an indicator (over 100 examples) and as they often amalgamate Christian and pagan iconography, they're a close indicator of Viking influence.
The major way in which the Vikings affected AS England is primarily through the succession issues after the death of Edgar. Aethelread pays them to go away but this simply encourages them to come back. There are more English coins in Scandanavia between 990-1040 than in England (check out Runestone U 344 from Orkesta in Sweden for a Scandanavian account). Aelthraed gives way to Sveinn of Denmark, Cnut succeeds Aethelraed's return from exile, etc., it's a horrible mess but shows that England steady.
What impact that has is hard to describe - should you allow the fact that the Viking succession (and failure) led to the Norman invasion (and the wholesale change of England after that)? I'd be tempted to leave things at a linguistic level - the language changes the most courtesy of the Vikings.
Huge. (This will be general since I'm on my phone, but my sources would be Bede's Ecclesiastical History of the English People and pretty much every annual entry in the Anglo-Saxon Chronicle from the mid-eight century on. I'm really just writing about how the Northmen and the events related to them were documented by the Anglo-Saxons.)
There were successive waves of Viking pillaging and settlement starting in the 8th century. Eventually, a large swath of land in northern and eastern England (primarily Wessex, IIRC) as noted by /u/flubb, I was mistaken in this was given to them outright in appeasement. This was called the Danelaw. Prior to this, the English kings were paying massive amounts of money to the Vikings in a ransom called the Danegeld.
In the early 11th century, the "viking" king Swegn and his son Cnut conquered the consolidated kingdom of England that was ruled by Aethelred Unraed. Swegn died unexpectedly, after Aethelelred had already abdicated the throne and fled to Normandy, leaving the country in chaos. Cnut leaves and Aethelred ends up coming back. Archbishop Wulfstan II writes about this time period in his work Sermo Lupi ad Anglos, which is a fascinating work of late Anglo Saxon homiletic literature. (c.f., my username. I can tell you more about Wulfstan and the Sermo Lupi than you'd even want to know! It just celebrated its 1000th anniversary this year, too!)
Cnut would later return and pick up where his father left off, conquering the nation. I would argue that this incursion and its effects: the exile of Edward ("the Confessor") in Normandy, the succession crisis of Cnut's and Emma's children, and the rise of the Godwins under Cnut's reign led to the fall of the Anglo-Saxon culture after the Norman invasion.