The backdrop to understanding this title is the Book of Isaiah. In Isaiah 44:6 God Almighty affirms: "I am the first and I am the last; apart from me there is no God." Again, in Isaiah 48:12, God said: "I am he; I am the first and I am the last," and God said this right after His pronouncement that "I will not yield my glory to another" (verse 11b). Christ's use of this title in Revelation 22:12-13 was thus undoubtedly intended to be taken as a claim to be God Almighty. No other conclusion is acceptable.
Of course, to the modern ear, the claim to be the Alpha and the Omega may seem strange. But for the ancient Jew, Christ was describing Himself in a way they would have readily understood. Though the letters Alpha and Omega are the first and last letters of the Greek alphabet, John recorded the Book of Revelation for Jewish readers who were also familiar with the Hebrew language and alphabet. And therein lies the significance of Christ's claim. In Jewish thinking, a reference to the first and last letters of an alphabet (aleph and tau in Hebrew) was regarded as including all the intermediate letters, and came to represent totality or entirety.
It is with this idea in mind that the Jews in their ancient commentaries on the Old Testament said that Adam transgressed the whole law from aleph to tau. Abraham, by contrast, observed the whole law from aleph to tau. The Jews also believed that when God brings blessing upon Israel, He does so abundantly, from aleph to tau.
When used of God (or Christ), the first and last letters express eternality and omnipotence. Christ's claim to be the Alpha and the Omega--like God Almighty's claim in the Old Testament--is an affirmation that He is the all-powerful One of eternity past and eternity future.