We don't have any specific information on Amdír's origins, but we do know that he was one of the Sindar (because his son Amroth is stated to have belonged to that people). He was therefore presumably part of the wave of Sindar who spread out eastwards across Middle-earth in the early Second Age, settling among and ruling the Silvan Elves who lived in regions such as Lórinand.
The interpretation 'looking up' comes from a source (Athrabeth Finrod ah Andreth in volume 10 of The History of Middle-earth) where the word amdír is explained as a philosophical concept related to hope rather than a personal name. Assuming the same interpretation applies to Amdír's name, it can presumably be interpreted as something like 'one who looks up (with hope)'.
The name Lórien did not come into use until long after Amdír's time, when Galadriel and Celeborn settled in that land. In his own time, Amdír would probably have been titled King or Lord of Lórinand, though those titles are not specifically recorded.