What is a god?
Gods are essentially immortal (or nearly immortal) supernatural beings with personal agency. The personal agency part is important, because otherwise we are speaking of ordinary natural phenomena or laws; it is their personal agency which distinguishes gods from the blind forces of nature.
(Gods are not necessarily completely immortal; being almost immortal is often good enogh. For example, Baldur of the Norse pantheon was famously killed by Loki — complicated story — with a spear made of mistletoe.)
As supernatural beings, gods cannot be fully comprehended by the minds of mortal men; various religions provide various methods of perceiving the divine, always through a form of empathy or ecstasy and never through reason. Some religions even state explicitly that mortal men cannot comprehend the divine and shouldn't even try.
But in practice, a god is simply a supernatural entity which is respected as a god by mortals. This purely phenomenological approach has the merit of avoiding complicated questions, such as, how can Venus be a goddess when Yahweh is the one and only God? Venus was respected as a goddess by millions of followers, had countless temples, priests and priestesses, she had rituals, she received prayers and sacrifices: this makes her a goddess, and any complicated questions regarding her relationship with the One of Buddhists or Gnostics, or with the one and only Yahweh, the God of Hebrews, Christians and Muslims can be left to learned theologians.
How can one become a god — the hard way
The luscious Greek mythology provides an excellent and detailed example of a mortal who became a very successful god, with thousands of temples and millions of devoted followers: Hercules, or to give him his Greek name, Heracles — "glory of Hera", a splendidly ironic name, given that Hera, the sister and wife of Zeus, was actually his determined antagonist.
The basic myth is that he was the son of Zeus, the supreme god, and Alcmene, a mortal woman. As the son of a mortal woman, Hercules was mortal; but Zeus wanted to provide him with divine powers, so he had his (and only his, no mother — long story) daughter Athena surreptiously place the infant at Hera's breast, so he could take divine milk. Athena did so, and Hera unknowingly nursed the infant Hercules, thus enabling him to gain superhuman powers; when she realised that the infant at her breast was the hated bastard son of her unfaithful husband, she threw him away, in the process spilling a few drops of milk on the firmament and producing the Galaxy, the "milk stain" (from galactos, "milk").
Hercules grew to be a superhero, performing amazing works, including descending into the netherworld and stealing the three-headed divine dog Cerberus who guarded the gates of Hell, taking part in the expedition of the Argonauts and liberating Prometheus from his chains.
Eventually he died, poisoned out of jealousy by his own wife Deianira with the blood of Nessus, a resentful centaur whom Heracles had killed. Heracles built his own funeral pire and climbed on it; as the pire was burning, Zeus raised him to Olympus and made him an immortal god.
After his apotheosis Heracles became the god of strength, heroes, sports, athletes, health, agriculture, fertility, trade, oracles and divine protector of mankind; many cities raised temples of Heracles and honored him as their protector god; there were at least ten cities named Heraclea (including modern Saint-Tropez, and three cities in Asia Minor which still keep the name — Ereğli in Turkish).
How can one become a god — the easy way
Become Roman emperor. Many emperors were made gods by their successors, with temples, and priests, and rituals, and sacrifices. Some even made themselves gods, although this was considered bad manners.
Become the lover of a Roman emperor. The best known example is Antinous, the lover of emperor Hadrian. After his death, Antinous was deified (and assimilated with Osiris, thus making Hadrian a lover of one of the highest gods of Egypt). The cult of Antinous proved exceptionally successful, persisting until the 4th century when it was finally suppressed by the increasingly fundamentalist Christian church.
Become a close relative of a Roman emperor. For example, Caligula deified his sister (and lover) Drusilla and raised temples in her honor, complete with priests, rituals and sacrifices.
Distinguishing between a creator and a god
This is the main concept behind neoplatonism and gnosticism. Neoplatonism, of which the best known exponent is Plotinus, was a religious and philosophical movement which started around the end of the 2nd or beginning of the 3rd century CE and for several centuries competed with mainline Christianism; gnosticism is a bit older and lasted a bit longer. The basic idea was that the world as we know it was created by a Demiurge (literally "craftsman" in Greek), a sort of lesser divinity of ambiguous ethic position who assembled the imperfect visible world; some gnostic strands even equated the Demiurge with the Devil. Only by transcending the impure world created by the Demiurge could a man hope to know (hence "gnosticism" from Greek gnosis, knowledge) the "real" world and be united with the One god.
(This is a grossly simplified and potentially misleading view. Do follow the provided links and read on neoplatism and gnosticism.)