To give a little more context to names in the 1st century, surnames of the type typical in English speaking countries today were not in use. Romans would often have multiple names, though not always. This was seen at best in the tria nomina, the three names, which would include a praenomen, that is a ‘first’ name, or a personal name; such a name was generally used by family and other intimates. Then there was the nomen, or nomen gentilicium, which would designate their family or clan (gens). A third name, the cognomen, would often identify an individual by a distinguishing feature, and in some instances functioned more like our ‘first’ names would.

But this is upper class Romans, which Jesus was not. We must instead talk about 1st century Judea and its context. Instead you get a typical pattern, also in relatively common use among Greeks, of a single name, and generally modified by identification of the father. So, in Greek, for instance, we might have Alexander son of Socrates. Indeed, the word [son] could be omitted. A similar practice is evident among the Greek-langauge texts of the New Testament, you have John son of Zebedee. In some instances, a person is named instead for the geographical hometown or another distinguishing feature.

In Jesus case, it’s very likely that he was referred to as Jesus son of Joseph. Or Yeshua ben Yosef, being the relevant construction in Hebrew. He would also be referred to as Jesus 'from Nazareth'.

Christ is not a surname at all, as others have noted. It is a title applied in the New Testament documents, equivalent to Messiah, and indicates the belief of the authors and of Jesus’ followers that he was the long-expected fulfilment of their interpretation of Old Testament prophecies about a saviour/royal figure who would appear in the last days to bring deliverance to the Jewish people and establish an eschatological kingdom. However, the consistent collocation of ‘Jesus’ with ‘Christ’ does pair the two terms to the extent that the force of its meaning may sometimes be less apparent. To explain more simply, if every time you say “Jesus” you actually say “Jesus Christ” the significance of “Christ” may be ‘lost’, it may simply be a speech tag that always goes with Jesus.

His last name wasn't "Christ." Interesting question, as I've often wondered that his last name was.

Think of "Christ" as a title. Many scholars think he was referred to as "Jesus the Christ."

Christ is the Greek word for "Messiah" actually, not a last name. Jesus Christ means "The Messiah Jesus".

When it comes to family names in general, usually Jewish people at the time of Jesus had a first name, and their last name was the first name of their father or the town they come from (that explains why Jesus is sometimes called "Jesus of Nazareth", or "Jesus of Nazareth son of Joseph" in the Bible).