"This is the message you heard from the beginning: We should love one another." NIV translation
John is reminding us of Jesus' command to love one another (see John 13:34). It is not a new message in that Jesus was constantly calling us to love and telling us to demonstrate that love by our actions.
"Do not be like Cain, who belonged to the evil one and murdered his brother. And why did he murder him? Because his own actions were evil and his brother's were righteous." NIV translation
Cain got jealous of his brother when God favored him because of his offering. This jealousy led to Cain killing his brother. The same type of thing happens today even among believers as some get jealous when they see God's favor on others. This can then lead to backbiting and assaults on their character. Instead, we should rejoice when we see God at work and showing favor to our brothers and sisters because it is all for the glory of God.
"Do not be surprised, my brothers, if the world hates you." NIV translation
If you are following Jesus, you will face opposition and even hatred. The good news of Jesus is foolishness to those that are perishing and so they will not understand you. These people will make fun of what they do not understand and in many cases will do everything they can to stop you.
"We know that we have passed from death to life, because we love our brothers. Anyone who does not love remains in death." NIV translation
The second part of our paternity test is the test of love. God is love and so, if we are His children, we will show love to others. If we are His children, we have life but if we are not then death still has a hold on us.
"Anyone who hates his brother is a murderer, and you know that no murderer has eternal life in him." NIV translation
Hate is the opposite of love and since we know that God is love then the opposite of God is death and hell. It is a tough lesson but there is no room for hate in the heart of a believer.