Bernard of Italy, illegitimate son of Pepin of Italy (himself a legitimate son of Charlemagne), became king of the Lombards in 810.
Edward the Martyr, briefly king of England from 975 to 978, was probably illegitimate; his father Edgar I acknowledged his younger son Æthelred as the only rightful heir (but Edgar's opinion lost most of its strength when he died).
Vladimir the Great became "Knyaz" of Kiev in 978, the title deriving from a Proto-Germanic word meaning "king". He was a natural son of Sviatoslav I, a previous ruler of the Kiev Rus'.
João I of Portugal was an illegitimate son of a previous king (Peter I) and conquered the throne after a civil war triggered by a lack of legitimate male heir.
Atahualpa was "illegitimate" son of Huayna Capac and still became Sapa Inca, roughly translated as "Emperor". The concept of illegitimacy is not the same as in contemporary Europe; the Inca had a primary wife (formally said to be his "sister"), whose children were heirs, and many secondary wives, who could be chosen for political reasons (hence not exactly "concubines" dedicated to the ruler's concupiscence). Atahualpa was son of one such secondary wife.
Paul I of Russia was most probably the son of Sergei Saltykov, and not of Peter III, husband of his mother Catherine (who became Catherine II). Paul succeeded Catherine (and was murdered 5 years later).