In Greek mythology there were at least 6 Alkaioi [Alcaei, "Alcaeus~es"], the best-known one being the one you're asking about: Amphitryon's dad. (The Athenian poet you've mentioned is one of several famous historical ["real-life"] figures bearing the same name.)
The mythical king Alkaios [Alcaeus] was one of the seven sons (perhaps the third-born of them) of the hero Perseus, a son of Zeus famous for having slain the Gorgon Medusa. Alkaios' mother, Perseus' wife Andromeda, was a princess of Aithiopia [Ethiopia], and herself a descendant of Zeus and Poseidon. Alkaios was probably born after his brothers Perses and Mestor, and his sister Gorgophone. Alkaios, Mestor and another brother, Sthenelos [Sthenelus], were married to daughters of Pelops. Alkaios' wife was called Astydameia, although there is an alternate version in which Alkaios is married to the Theban princess Hipponome. In yet one other version Alkaios' wife was Laonome, the daughter of an otherwise unknown Arcadian named Guneus.
Pittheus, a son of Pelops and therefore a brother-in-law of Alkaios, founded the city of Troizenos [Troezen] in the territory ruled by the sons of Perseus. Alkaios became a co-ruler of the city together with Pittheus, who would later become grandfather of Theseus, who would gain fame for killing the Minotaur.
Alkaios had a daughter Anaxo, therefore sister of Amphitryon, who he married to his own brother, her uncle Elektryon [Electryon]. Anaxo and Elektryon became the parents of Alkmene [Alcmene], who grew up to marry Amphitryon, who was thus both her cousin and her uncle. Alkmene bore Amphitryon a son and a daughter, and she also was the mother by Zeus of the greatest Greek hero Herakles [Heracles], whom the Romans called Hercules. Alkaios is also said to have had a third child, a daughter named Perimede, who married her own cousin Likymnios [Licymnius], the illegitimate son of Elektryon by a slave named Mideia.
When Herakles was born, Amphitryon named him Alkaios, after Amphitryon's own father, or he called him Alkides [Alcides], "[Son] Of Alkaios." Herakles later changed his name after he had, in a fit of madness, killed his own wife and children, and was attempting to appease his stepmother the goddess Hera by calling himself "Hera's Glory," which is what Herakles means.
Herakles himself had a son named Alkaios, whom he had sired upon a Lydian slave, and whose descendants were kings of Lydia. This Alkaios is otherwise said to have been called Kleolaos [Cleolaus].
Another Alkaios was a king of Paros who was involved in a misadventure with Herakles. For details on him and another mythical Alkaios who fought in the Trojan War, see here:
https://en.wikipedia.org/wiki/Alcaeus_(mythology)
For more on Amphitryon, check this out:
http://www.maicar.com/GML/Amphitryon.html
