It seems that Jesus is what the New Testament tends to consist of; him and his teachings.
Did he ever ‘teach’ about the Bible, the first testament, in any capacity?
Although Jesus didn't write about the Bible (which at that time would of course be just the Old Testament), He did teach about it. In addition to the specific quotations listed in Ruminator's answer, there's Luke 24:27 about the discussion on the road to Emmaus:
Beginning at Moses and all the prophets, he expounded to them in all the scriptures the things that were concerning him.
And he came to Nazareth, where he had been brought up: and, as his custom was, he went into the synagogue on the sabbath day, and stood up for to read.
Luke 4:16 KJV.
Luke records that it was Jesus' custom to attend synagogue on the sabbath in the town where he had been brought up by Joseph and Mary, and to read from the scripture in public.
Once baptised (by John the Baptist) and after being tried with temptation, he came again to Nazareth, where, following his usual custom, he then read and, this time, spoke from the passage Isaiah 61: 1-2.
His audience "rose up, and thrust him out of the city, and led him unto the brow of the hill whereon their city was built, that they might cast him down headlong."
But he passing through the midst of them went his way. Luke 4:30.
The only recorded instance of Jesus writing anything that I'm aware of is this account which has an uncertain provenance:
[Jhn 8:2-11 NKJV] 2 Now early in the morning He came again into the temple, and all the people came to Him; and He sat down and taught them. 3 Then the scribes and Pharisees brought to Him a woman caught in adultery. And when they had set her in the midst, 4 they said to Him, "Teacher, this woman was caught in adultery, in the very act. 5 "Now Moses, in the law, commanded us that such should be stoned. But what do You say?" 6 This they said, testing Him, that they might have [something] of which to accuse Him. But Jesus stooped down and wrote on the ground with [His] finger, as though He did not hear. 7 So when they continued asking Him, He raised Himself up and said to them, "He who is without sin among you, let him throw a stone at her first." 8 And again He stooped down and wrote on the ground. 9 Then those who heard [it], being convicted by [their] conscience, went out one by one, beginning with the oldest [even] to the last. And Jesus was left alone, and the woman standing in the midst. 10 When Jesus had raised Himself up and saw no one but the woman, He said to her, "Woman, where are those accusers of yours? Has no one condemned you?" 11 She said, "No one, Lord." And Jesus said to her, "Neither do I condemn you; go and sin no more."
These are some of the many sayings recorded in the gospels where Jesus said "It is written":
[Mat 4:4, 7, 10 NKJV] 4 But He answered and said, "It is written, 'Man shall not live by bread alone, but by every word that proceeds from the mouth of God.' " ... 7 Jesus said to him, "It is written again, 'You shall not tempt the LORD your God.' " ... 10 Then Jesus said to him, "Away with you, Satan! For it is written, 'You shall worship the LORD your God, and Him only you shall serve.' "
[Mat 21:13 NKJV] 13 And He said to them, "It is written, 'My house shall be called a house of prayer,' but you have made it a 'den of thieves.' "
[Mat 26:24, 31 NKJV] 24 "The Son of Man indeed goes just as it is written of Him, but woe to that man by whom the Son of Man is betrayed! It would have been good for that man if he had not been born." ... 31 Then Jesus said to them, "All of you will be made to stumble because of Me this night, for it is written: 'I will strike the Shepherd, And the sheep of the flock will be scattered.'
[Mar 7:6 NKJV] 6 He answered and said to them, "Well did Isaiah prophesy of you hypocrites, as it is written: 'This people honors Me with [their] lips, But their heart is far from Me.
[Mar 9:12-13 NKJV] 12 Then He answered and told them, "Indeed, Elijah is coming first and restores all things. And how is it written concerning the Son of Man, that He must suffer many things and be treated with contempt? 13 "But I say to you that Elijah has also come, and they did to him whatever they wished, as it is written of him."
[Mar 11:17 NKJV] 17 Then He taught, saying to them, "Is it not written, 'My house shall be called a house of prayer for all nations'? But you have made it a 'den of thieves.' "
[Mar 14:21, 27 NKJV] 21 "The Son of Man indeed goes just as it is written of Him, but woe to that man by whom the Son of Man is betrayed! It would have been good for that man if he had never been born." ... 27 Then Jesus said to them, "All of you will be made to stumble because of Me this night, for it is written: 'I will strike the Shepherd, And the sheep will be scattered.'
Below is a popular poignant poem about this sandaled Messiah, celebrating his lack of secular and religious trappings and credentials:
This is a popular poem about the life of Jesus Christ. Although the author is frequently cited as "unknown" the poem is actually attributed to James Allen Francis.
*> He was born in an obscure village, the child of a peasant. He grew up
in another village, where he worked in a carpenter shop until he was 30. Then, for three years, he was an itinerant preacher.
He never wrote a book. He never held an office. He never had a family or owned a home. He didn't go to college. He never lived in a big city. He never traveled 200 miles from the place where he was born. He did none of the things that usually accompany greatness. He had no credentials but himself.
He was only 33 when the tide of public opinion turned against him. His friends ran away. One of them denied him. He was turned over to his enemies and went through the mockery of a trial. He was nailed to a cross between two thieves. While he was dying, his executioners gambled for his garments, the only property he had on earth. When he was dead, he was laid in a borrowed grave, through the pity of a friend.
Twenty centuries have come and gone, and today he is the central figure of the human race. I am well within the mark when I say that all the armies that ever marched, all the navies that ever sailed, all the parliaments that ever sat, all the kings that ever reigned--put together--have not affected the life of man on this earth as much as that one, solitary life.*
*Attributed to James Allen Francis.
He did not have a written commentary at all. But as Jesus was a Jewish boy who grew up in Jewish culture, every young Jewish child grew up familiar with the Law and the Prophets from at least 2 years old until He reached 13 (adulthood back in their day). This was why when He visited the Temple (when He stayed behind) boy Jesus amazed even temple teachers who were definitely decades older than He was.
Prior to His public ministry as well, Jesus underwent a further study of the Torah, Prophets, and other culturally Jewish divine texts for - approximately - another 18 years as in Jewish culture the proper age to publicly become a rabbi was the moment you hit 30, the age of maturity.
Yes He did teach the Law & The Prophets in capacity, stating in Matthew 5 that He came to fulfill it and not abolish it (v. 17). He even summarized the ENTIRE narrative of the Old Testament by quoting the two greatest commandments: "Loving God first and loving others as you love yourself."