There definitely were contemporary accounts - once. We know that several people were writing about his campaigns as they happened, including his general Ptolemy, the later founder of the Ptolemaic dynasty in Egypt. However, all these works are lost. The surviving accounts of Alexander's life were written several centuries later. The earliest is Diodoros of Sicily (1st century BC); then there are a bunch that date to the Principate (Curtius Rufus, Plutarch, Arrian). It is not always easy to tell where these authors got their information, how much of it was distorted over time, and how these historians' own historical context influenced their writings. Somewhat counter-intuitively, the latest of the comprehensive accounts that survive to our time (Arrian's Anabasis) is usually considered the most reliable.

This is not to say, though, that there is no contemporary evidence for Alexander the Great. We have his inscriptions and coins; the mole he built to approach the walls of Tyre still attaches that ancient site to the mainland; the ruins of the palace he burned at Persepolis are there for us to see.