Matthew was likely written for Greek-speaking Christians of Jewish descent.
In my answer to your first question, I discussed reasons for believing Matthew was written in Greek rather than Hebrew. And it should be self-evident that the gospels were written to Christians. So here I'm going to list the reasons for believing the intended audience was specifically Christians of Jewish descent.
The Gospel of Matthew, more than the other gospels, often concludes a passage by introducing a Jewish Scripture and claiming Jesus has now fulfilled it.
In many cases, Matthew is simply drawing a parallel between events from the life of Jesus and events from Israel's history. Examples are Matthew 1:22-23 quoting a passage about a sign given to King Ahaz, Matthew 2:15 quoting a passage about the exodus from Egypt, and Matthew 12:17-21 quoting a passage about God's servant Israel (see Isaiah 41:8-9 for identification of the servant with Israel).
Structure of the Gospel
Just as Matthew draws parallels between Jesus' experiences and those of Israel, the gospel itself parallels the Torah, the five books of Moses. In Matthew, Jesus' teachings are combined into five long discourses.
- Sermon on the Mount - Matthew 5-7
- Missionary Discourse - Matthew 10
- Parabolic Discourse - Matthew 13
- Discourse on the Church - Matthew 18
- Olivet Discourse - Matthew 24
Each of these discourses contains more uninterrupted teaching than the parallel sections of Mark or Luke.
Respect for Jewish law and customs
The phrase "kingdom of God" appears 62 times in the New Testament, but Matthew uses the substitute phrase "kingdom of heaven" 31 times, even where the parallel accounts in other gospels say "kingdom of God". This is likely due to a Jewish custom of never writing the word "God" as a precaution against using God's name in vain.
In Jesus' teaching about handwashing, Mark 7:19 indicates that Jesus thereby "declared all foods clean".
He said to them, "Then do you also fail to understand? Do you not see that whatever goes into a person from outside cannot defile, since it enters, not the heart but the stomach, and goes out into the sewer?" (Thus he declared all foods clean.)
But in the parallel in Matthew 15, this parenthetical comment is omitted. We know from Paul's letters that some Jewish Christians kept part of their unique dietary restrictions as a matter of conscience. Matthew perhaps did not want to offend their sensibilities.
In Mark 10 Jesus prohibits all divorce. However, in the parallel in Matthew 19, Jesus makes an exception for marital unfaithfulness. The version found in Matthew acknowledges that divorce is allowed in Deuteronomy under certain circumstances.
Furthermore, Jesus says in Matthew 5:17, in a statement not found in the other gospels, that he has come not to abolish the law but to fulfill it. (Fulfill is probably used here in the same sense as above.)
Matthew's attention to the Jewish scriptures and respect for Jewish customs, as well as the structure of this gospel, indicate that it was written primarily for Jewish Christians rather than pagan converts.