Outside of the New Testament and writings of the Early Church there is some evidence, but it is not plentiful. It consists of historical references made 50-100 years after Jesus' lifetime. The earliest Gospel was written about 30 years after his lifetime. Unfortunately, it is all too easy to find arguments in books and on the internet from Christians and atheists that try to either overstate or understate their importance.
The writings are:
- Flavius Josephus - Speaks of Christ and Christians. The reference is authentic and the writings also mention John the Baptist and James, the brother of Jesus. The text often cited by some Christian apologists has been called into question as far as the original content and what had been inserted later. The sciences of textual criticism and higher criticism were used to identify the insertions based on style, linguistic compatibility, and vocabulary.
- Pliny the Younger - discusses Christians as a group, not Jesus.
- Suetonius - discusses Christians and a possible reference to Jesus.
- Tacitus - mentions death of Jesus and Christians
The debate rages on about what these texts prove or don't prove about Jesus. My conclusion is that these texts are viable sources indicating a man named Jesus existed and that he had a group of followers soon after his death. They are not proof of the supernatural claims about Jesus made in the New Testament or subsequent Christian writings.
Out of fairness to historical context, there were no daily newspapers at that time and most of what was written pertained to matters of government, history, or religion. The literacy rate was very low. Most of the surviving writings were from the Roman government and Roman and Jewish historians. It would not be expected that there would exist a lot of public references to an itinerant, apocalyptic preacher with a small following who was reported to perform miracles and persecuted by the Romans, as such preachers and prophets were not uncommon in that period.
I recommend the book Did Jesus Exist? by Bart Ehrman, a New Testament scholar, historian, and agnostic, particularly Chapter Two, Non-Christian Sources for the Life of Jesus. He is qualified to speak to the subject from a scholarly viewpoint, though his position has angered both Christians and also some atheists who insist that there is no evidence that the historical Jesus existed. All in all, it is a good read, especially if you are not familiar with how scholars examine religious texts and historical writings to ascertain their authenticity, date of writing, and significance.