"'By examining him yourself you will be able to learn the truth about all of these charges we are bringing against him.' The Jews joined in the accusation, asserting that these things were true." NIV translation
The prosecutor summed up his case against Paul and all of the Jews agreed with him. He tries to make it seem like even a child could see it in hopes of getting a quick agreement.
"When the governor motioned for him to speak, Paul replied: 'I know that for a number of years you have been a judge over this nation; so I gladly make my defense.'" NIV translation
Now, Paul gets to defend himself against these false accusations. He starts by saying that he knows that Felix understands the Jewish customs and so he is glad to present his defense. Felix had been there for years dealing with Jewish matters and so would be qualified to make a reasonable ruling in the case.
"'You can easily verify that no more than twelve days ago I went up to Jerusalem to worship. My accusers did not find me arguing with anyone at the temple, or stirring up a crowd in the synagogues or anywhere else in the city. And they cannot prove to you the charges they are now making against me.'" NIV translation
Paul continues by explaining that he only went to worship and that his accusers weren't even there to see what he was doing. They have no basis for their charges and cannot prove them. It is interesting to note that he says it was "no more than twelve days ago" that this happened. Twelve is the number of divine authority throughout the Bible and it reminds us that Paul is in front of Felix by the appointment and authority of God and so he has no need to worry.
"'However, I admit that I worship the God of our fathers as a follower of the Way, which they call a sect. I believe everything that agrees with the Law and is written in the Prophets,'" NIV translation
Paul continues by explaining that He is a follower of Jesus Christ which is in agreement with the Torah (Old Testament) which his accusers believed. Jesus said that He did not come to abolish the Law but to fulfill it and all of the Old Testament points to the coming of the Messiah who is Jesus, the Son of God.
"and I have the same hope in God as these men, that there will be a resurrection of both the righteous and the wicked. So I strive always to keep my conscience clear before God and man." NIV translation
Paul explains that there is one hope and that is for the resurrection just as Jesus was raised from the dead. In light of that hope, he tried to live a life that would bring honor and glory to God. He gave practical advice on how we can all do this in Ephesians 4.