What does King mean?
The English vocabulary word 'king' (Old English 'cyning') used directly as a given name. A transferred surname/word-name rather than an etymological 'king' name from another root.
What does Remington mean?
An English habitational surname from Rimington (Yorkshire/Lancashire), from Old English 'Riming' (boundary stream) + 'tun' (settlement, enclosure), i.e. 'settlement on the boundary stream'. Often loosely glossed 'place on a riverbank'; it also carries firearm-brand associations.