Jesus Christ spoke Aramaic. In Mark 5:41, we see Jesus saying "Talitha Cumi." This is Aramaic.
Aramaic was the language of first century Israel. Not Greek. Many scholars try to give the false impression that Greek was the language of first century Israel.
Let me give you the historical evidences from Josephus.
Jewish Historian Josephus wrote:
"I have also taken a great deal of pains to obtain the learning of the Greeks, and understand the elements of the Greek language, although I have so long accustomed myself to speak our own tongue, that I cannot pronounce Greek with sufficient exactness; for our nation does not encourage those that learn the languages of many nations, and so adorn their discourses with the smoothness of their periods; because they look upon this sort of accomplishment as common, not only to all sorts of free-men, but to as many of the servants as please to learn them. But they give him the testimony of being a wise man who is fully acquainted with our laws, and is able to interpret their meaning; on which account, as there have been many who have done their endeavors with great patience to obtain this learning, there have yet hardly been so many as two or three that have succeeded therein, who were immediately well rewarded for their pains." - Antiquities of Jews XX, XI
Jewish Wars (Book 1, Preface, Paragraph 1) - "I have proposed to myself, for the sake of such as live under the government of the Romans, to translate those books into the Greek tongue, which I formerly composed in the language of our country, and sent to the Upper Barbarians. Joseph, the son of Matthias, by birth a Hebrew, a priest also, and one who at first fought against the Romans myself, and was forced to be present at what was done afterwards, [am the author of this work]."
In Antiquities of Jews Book 3, Josephus points out that Hebrews called Pentecost "Asartha." Asartha is Aramaic, because Aramaic places the definite article at the end of the word, thus the 'tha' at the end of 'Asartha' is the Aramaic definite article on a feminine noun. This is the same thing with the Aramaic word Talitha (Mark 5:41).
Unlike Aramaic, the definite article of Hebrew is in the beginning of the word ("Ha"). If Josephus was writing Hebrew, then the word "Asartha" would have become "Ha Atzeret."
Like Hebrew, we use the definite article ("the") in the beginning of a word in English. For example, we say "the car" in English. We never say "car the."
Let me give you another example.
John 19:17 (KJV) - "And he bearing his cross went forth into a place called the place of a skull, which is called in the Hebrew Golgotha:"
When they say "Hebrew", they are referring to their Hebrew tongue which was Aramaic in first century AD. Notice the word Golgotha. This is Aramaic, because "tha" in Golgotha is Aramaic definite article on a feminine noun.
That is why NIV, ESV, and other bible versions write "Golgotha, Gabbatha, etc." as Aramaic instead of Hebrew.
John 19:17 (NIV) - "Carrying his own cross, he went out to the place of the Skull (which in Aramaic is called Golgotha)."
If I write Aramaic word "Golgotha" in Hebrew, then the word will become "Ha Gulgoleth."
In first century AD, Jews called Aramaic "Hebrew", because that is the language of Hebrews. Jews are also called Hebrews, because they are the descendants of Abraham the Hebrew (Genesis 14:13, Philippians 3). Even today, Many Iraqi Jews call their Aramaic "Hebrew" ("Ibraith" in Aramaic), because it is the language of Hebrews. We call Deutsch "German", because it is the language of German People.
Another point that should be noted is Old Hebrew was preserved through scrolls in first century AD, because it is considered as the holy language among Jews. High Priests preserved Hebrew scrolls for religious purposes in temple of Jerusalem.
But the spoken language of Israel was Aramaic (Gabbatha, Golgotha, Asartha, etc.) in first century AD. Just like Jews preserved Old Hebrew for religious purposes in first century AD, many Hindus preserve Sanskrit in Kerala (a state of India) for religious purposes, because it is the holy language of Hindus. But the spoken language in Kerala is Malayalam.
Judean Aramaic was also known as Hebrew in order to differentiate the way Aramaic is spoken in Judea and Aramaic spoken in Galilee and Syrian regions.
Through Matthew 26:73 and Mark 14:70, Peter was exposed by his Galilean Aramaic speech among people. Judeans used Dead Scrolls Alphabet to write Aramaic while Syrians commonly used Estrangela Alphabet to write Aramaic in first century AD. The Galilean accent of Aramaic would have sounded to the Judean Aramaic somewhat like Cockney sounds to a British aristocrat. In Talmud, Galileans are also ridiculed for the way they spoke Aramaic.
Aramaic became known as Syriac, because First century Greek Historian and Geographer Strabo points out that Greeks called Arameans "Syrians" in his book "Geography." Unlike Jews in Judea, Arameans called Aramaic which became known as "Syriac" or "Suristi" in Greek.
Also note the names in English Bible of New Testament - "Bar"tholomew, "Bar"abbas, "Bar"nabbas, "Bar"sabbas, "Bar" Jesus, Simon "Bar" Jonah, "Bar" Timaeus, etc.
Aramaic word Bar means Son. In Hebrew, Ben means Son. Even Rabbis point out that "Bar" in Bar Mitzvah comes from Aramaic.
In Josephus' Jewish Wars, one of the leaders who fought against Romans was Simon Bar Giora. Bar Giora means "Son of a proselyte" in Aramaic. Peshitta Tanakh is first century Old Testament written in Aramaic.
Also here are couple of Aramaic words found in Greek NT manuscripts - Satana (Luke 10:18), Mammona (Luke 16:9), Khqel Dama (transliterated as Akel dama in Greek in Acts 1:19), Maran Atha (1 Corinthians 16:22), Golgotha (John 19:17), Talitha (Mark 5:41), and Lebontha (Matthew 2:11)." So we even see Aramaic words in Greek NT manuscripts.
Below is how we write above Aramaic words in Hebrew.
Satana (Luke 10:18) - In Hebrew, the word "Satana" will become "Ha Satan."
Mammona (Luke 16:9) - In Hebrew, the word will become "Ha Mammon."
Khqel Dama (transliterated as Akel dama in Greek) - In Hebrew, "Sh'deh Hadam."
Maran Atha (1 Corinthians 16:22) - In Hebrew, "Adonainu Atha."
Golgotha (John 19:17) - In Hebrew, "Ha Gulgoleth."
Talitha (Mark 5:41) - In Hebrew, "Ha Yaldah."
Lebontha (Matthew 2:11) - In Hebrew, "Ha Lebonah."