Alfred's patrilineal heritage is only really certain as far back as his grandfather Ecgberht, who ruled Wessex from 802 until his death in 839. Alfred's biographers would have us believe that he is a Gewisse, a member of the Cerdicing dynasty with a clear path back to Cerdic, the very first king of Wessex, albeit through a 'junior' line that deviates via Ine through the kingdom of Kent. This could be the case, although Ecgberht's origins are muddy at best and his family remains the source of great debate; presuming that he is descended from Ingild, brother of Ine of Wessex, that heritage is itself not entirely precise. The dynastic connection appears to have been important to Alfred's immediate family: his brother Æthelred was laid to rest in the Minster church at Wimborne founded by his great (great great...) aunt Cuthburga, and it was also there that his son Edward confronted his nephew Æthelwold over the West Saxon succession in 899. Alfred's matrilineal history is much more obscure: his mother was Osburh, the daughter of one of Æthelwulf's closest household, Oslac and possibly descended from the Ealdormen of the Isle of Wight.

It's in this early Cerdicing history that we have some potential for Roman descent. Much debate has been had about the extent to which the West Saxon dynasty was formed from the integration of Germanic and Sub-Roman Brythonic families, as reflected in a number of names. 'Cerdic' itself is often said to be more of a Brythonic name than a Germanic one (see Susan Oosthuizen's The Emergence of the English for a modern review of this debate), and Ine's predecessor (and cousin of... some sort...) Cædwalla has an undeniably Brythonic name. What this means is that if Alfred was indeed definitely a descendant of Ingwold and also therefore of Cerdic, he is likely to have had at least some Brythonic- and thus potentially Romano-British - heritage.

As to the second part of your question, I wrote an answer about the importance of Romanitas to Eary Medieval English politics here and a more recent post in our Vikings AMA here.