Alfred the Great (he of the burnt cakes) is generally taken to be the first King of England. His grandson Aethelstan was the first to claim that title, but who's heard of Aethelstan? Nobody, so we'll say Alfred. It all works out the same anyway.
The line down from Alfred was broken a couple of times by the Vikings, but managed to make it through nearly to the Norman Conquest in 1066. Harold Godwinson (he of the arrow in the eye) was not the closest heir to Edward the Confessor - that honour went to Edward the Exile, the Confessor's nephew.
Obviously, the Conquest broke the chain of Saxon kings - but it's possible to trace their lineage quite a lot further. (In the tree below, the hypothetical line of succession is numbered. Historical monarchs are bolded; other members of the hypothetical line are italicised. Only children and spouses relevant to the succession are listed.)