Perhaps the most famous instance of foreign claims being made to the English crown was in 1066: William the Conqourer (Norman) Harold Hadrader [and Magnus] (Norweigen) Sweyn Estridson (Danish)
A later examples which springs to mind is: William of Orange (Dutch)
Now I concede that issues surrounding national identity through much of the Middle Ages aren't comparable to modern notions. Nonetheless for a bit of fun I'm keen to understand who else during what we might broadly define as the Feudal period who might be perceived as 'foreign' had a claim/claimed the Kingdom of England?
Presumably the Scots at some point? Did any of the HRE provinces do so? I understand that later the Russian and German monarchies were closely related to the English throne. Did this extend to any direct claims during this earlier period?
Louis of France (Louis the Lion), the son of Philippe Augustus, claimed England from King John in 1216. It wasn't a particularly strong claim as it was through his wife - Blanche of Castile, the granddaughter of Henry II (through her mother) - but the English barons (who often had possessions on the French side of the channel for which they paid homage to Philippe Augustus) were fighting John already and invited Louis in. As soon as Louis invaded in the south, Alexander II of Scotland invaded the north, with the aim of reclaiming some of his English possessions and sealing the deal by doing homage to Louis for them. It didn't work out. The Pope excommunicated Louis (he'd previously excommunicated John), the English barons started to desert him, and the wars seem to have ended more or less in a stalemate. John died during this period, but he left an heir; Louis gave up his claim at the Treaty of Lambeth; and Alexander married one of John's daughters. It might have turned out rather differently if John had left no sons. It's unlikely the French could have held England for long, but it might have been an interesting situation, especially as Louis only reigned in France for 6 years.