Did anyone write about Jesus at the time at which he was alive?
I can't find any sources from the time in which he was alive after a quick preliminary search.
No, nobody did. Despite the fact that literacy was relatively high amongst the Jews at that time. And we have several historians living there or in the area in those days. Jesus was mentioned outside the gospels a few times, but those lines are generally seen as either pious forgeries or out of context.
Josephus mentions Jesus, but it doesn't fit in the text and contradicts Josephus own religious ideas.
Pliny the Younger mentions Christians, as followers of a certain Crestos or Christos. He doesn't go into the existence of Jesus, merely asking the emperor what to do with his followers. Pliny lived a century later, by that time there were Christians. It doesn't prove Jesus lived, it proved Christians existed.
Now, since we're in the realm of religion here, I always wondered why nobody outside the bible mentions that the dead walked the streets of Jerusalem upon the death of Jesus. (Matt 27:50-54) You'd think this was something rather remarkable, worthy enough to write down for posterity or inform friends about. But nobody did. At least the governor should have informed his emperor about this rather unusual event. But even he didn't.
None that we know of. And actually Jesus wasn't unique in that regard. There were a lot of ancient figures who we know existed, but nearly everything we have written about them was from after their deaths (Alexander the Great, for example).
I'm going to guess, though, that what you're really asking is whether or not Jesus existed. The short answer is yes. He did! Very few historians worth their salt would question that, Christian or otherwise. Evidence that someone named Jesus lived during the First Century in Judea, was crucified, and was the central figure of Christianity is so rampant as to be virtually indisputable. This evidence includes the written account of Tacitus, who mentions Jesus and who was so hostile towards Christians that his passage is almost certainly not a Christian alteration of the original text. Furthermore, the existence of Jesus is supported by models such as the Criterion of Multiple Attestation and the Criterion of Embarrassment.
If the question is did anybody write about Jesus before his crucifixion then the answer is No.
If the question is are there Contemporary sources who wrote about Jesus then the answer is Yes.
While no sources are known who wrote of Jesus before the crucifixion there are writings by a contemporary of Jesus (4,6BCE–30CE) who talked with and wrote of people who knew Jesus. Also Josephus is not dismissed by most scholars as an independent verification of Jesus’s life.
Contemporary means living at the same time. The oldest books of the New Testament Romans, Corinthians, and Galatians take the form of letters from Saint Paul (4BCE–62CE) to early church leaders (dated 55, 56, 57 CE). And stories about St Paul’s experiences. St Paul was a contemporary of Jesus being about the same age as Jesus. Although Paul states he had never met Jesus before the crucifixion. He discusses meeting in his letters with St. Peter who was one of the apostles who knew and walked with Jesus.
The earliest letters of St. Paul predate Josephus by about 40 -50 years. But Josephus also contain two references to Jesus (books 18, 20) only one of which is contested. The contested verse is not thought to be entirely false insertion but to have been modified. The other verse from book 20 is not questioned for its authenticity.
Josephus on Jesus
Modern scholarship has largely acknowledged the authenticity of the reference in Book 20, Chapter 9, 1 of the Antiquities to "the brother of Jesus, who was called Christ, whose name was James" and considers it as having the highest level of authenticity among the references of Josephus to Christianity.
Mainly the Deuterocanonical books of the Old Testament:
(Prior to Paul most of the old testament had been translated into Greek (mid-3rd BCE), a language consumable to non Jewish Roman Citizens.).
The first part of Joseph Klausner's Jesus of Nazareth: His Life, Times, and Teaching contains a rather thorough discussion of the existing sources, even though it has to be noted, that the book is several decades old, and some recent archeological findings or research are not reflected there (e.g., it dismisses tegerences to Jesus in Josephus' writings, mentioned in another answer).
More specifically, the first three books of the New Testament are retellings of the same Hebrew or Aramaic source, likely written by a Jesus' contemporary.
Jesus is also mentioned in many contemporary Jewish writings (Talmuds, etc.), although he often appears there under different names, so establishing his identity requires historical research. (And he is obviously not treated as a son of the Almighty, but as a sage/rabbi.)
Klausner's overall conclusion is that we have more evidence of Jesus' existence, than we have about Socrates, even though no one doubts the existence of the latter.
As a lot of people have said, there are many people who were mostly only written about after their death in the ancient world. However, there were people who wrote about Jesus around the time of his existence and his execution that were not Christian like Roman historian & senator Tacitus. One of the passage of his work Annals talks about the execution of Jesus:
"called Christians by the populace. Christus, from whom the name had its origin, suffered the extreme penalty during the reign of Tiberius at the hands of one of our procurators, Pontius Pilatus."
Scholars generally say that it is authentic since many of the events mentioned in the book also match up with records of other events at the time. Historian William L. Portier even stated that there was consistency between the references by Tactitus, Josephus (a Jewish historian who wrote about Jesus), and letter from Pliny the Younger to the Emperor Trajan (where he wrote about "Christus", which makes since many historians like Charles Guignebert don't doubt that Jesus lived in or or near Gallilee in the first century). So historians generally agree with evidence like this that Jesus probably existed and wasn't just a myth: it is just simply a possibly that a popular Jewish teacher who was executed as a heretic was mythologized (which was pretty common in the era and even in times near common day, like how George Washington has been worshipped as god by Hawaiian Shintoists for years even though most of us see him as just a man & how the inventor of the language Esperanto L. L. Zamenhof was made a spiritual deity in the Oomoto religion).
Heres the josephus passage from an Arabic translation from the 10th century, a lot more neutral. “At this time there was a wise man who was called Jesus. And his conduct was good, and [he] was known to be virtuous. And many people from among the Jews and the other nations became his disciples. Pilate condemned him to be crucified and to die. And those who had become his disciples did not abandon his discipleship. They reported that he had appeared to them three days after his crucifixion and that he was alive. Accordingly, he was perhaps the Messiah concerning whom the prophets have recounted wonders.” (10th Century Arabic Text)