Abraham and David's Justification
When speaking to Jewish believers, there wasn't anyone that Paul could reference who was more important to their heritage than Abraham. All Jews looked to Abraham as their common ancestor. It was his children, after all, who defined the tribes of Israel through their descendants.
So in Romans 4:1-5, Paul talks first about what it was about Abraham that caused him to be considered righteous by God and to be chosen by Him for the great task that God set before him. He could not have been righteous by following the letter of the law, because the law wouldn't be given for another 430 years after God chose Abraham. It wasn't the circumcision that saved Abraham. because at the time he was chose, he was uncircumcised.
In fact, at the time of God choosing Abraham, he would have been considered a heathen in Jewish eyes. But God looked on Abraham and saw his belief in Him, and by grace counted it for righteousness (Genesis 15:6). It is by this same basis that all who were not Jewish had access to God. Our belief in the salvation plan that God has provided us and our belief in God allows that same grace to work in our lives for salvation. It is that same faith that God saw in Abraham that allows God's grace to work for us, just like Abraham.
Romans 4:6-8 goes on to extend this example to David. Quoting from Psalms 32:1-2, Paul declares that even David, a man God said was after His own heart, and was under the law, would have been judged by that law. He was blessed because he sought God's forgiveness when he sinned, and once he had gotten right with God,. his past sins were forgiven and would therefore not be imputed to him. His sins were covered, then by the blood of the animal sacrifice, and ultimately by the blood of Jesus.
But, like Abraham, he had to choose to follow God's leading in his life. Being a king, and even a famous king, didn't make his sin relationship with God any different than any other Jew or Gentile. If we die in sin, that sin isn't covered and will be imputed to us by God. It is clear from passages like this that a concept of once saved, always saved is false. You can read the 210 Plain Laws and Warnings article by Finis Jennings Dake for additional scriptures which make this clear.
The Law and Faith
Romans 4:9-12 then address the time frame for when this righteousness was imputed to Abraham, and concludes that it was done while he was still uncircumcised showing that the physical act for males of becoming a Jew wasn't important for Gentiles. Faith alone is needed to get God's grace.
He concludes this section by telling his readers that the promise that God made to Abraham was a promise made not under the law, which didn't formally exist yet, but was imputed by the righteousness of Abraham through his faith in God.
Romans 4:13-16 lets the readers know that the promise of God to Abraham that he was the heir to the world was not limited to just those covered under the law, but was also available to all those who, like Abraham, were counted worthy by their faith in the blood of Jesus. The law itself only declared sin and provided a temporary method of covering that sin. If the steps weren't taken, the penalty of sin was on whoever committed it. Just as the Law dealt with sin in the Jewish faith, so it is with us today. The Law is still there to condemn those who sin against God. But we have a means of salvation through Christ's blood to obtain mercy through the grace of God and our faith.
Abraham's Faith with Sarah
Finally, in Romans 4:17-25, Paul describes the faith of Abraham who was around 100 years old when God told him he would be a father of more descendants than could be numbered. At that age, and considering the age of Sarah his wife, he could have been filled with doubt. Sarah, in fact, laughed at the idea of her being a mother at her age.
But Abraham had faith and believed God's promise and indeed fathered what would become the Jewish race. He trusted in the promise of God and had faith. That faith was not only for those Jewish people who would follow, but also all those who would likewise share in God's salvation through their faith in God.
I'd like to also add just a bit of an aside in this section of the chapter to address some other concerns that we have. God has made many promises to His people. God can be counted on to keep His word at all times. But we also need to do our part. God gave Abraham and Sarah, what at their ages must have seemed like an impossible, and indeed laughable promise. But Abraham had faith, and God fulfilled His promise.
Many readers are facing huge troubles in their lives today. Perhaps your health is failing. Perhaps you have financial needs that you don't know how to get past. Perhaps you have relationship issues whether with your wife or husband, children, other family members, at work or school, with neighbors, or maybe you very much want the right relationship, but it hasn't materialized. God's word is full of promises to meet your needs.
But we also need to have the faith of Abraham that God will work out the situations in our lives the best He can. We also have to trust that God wants the best outcome for each of us. Not everyone who prays for healing is healed. I think that more would be healed if our faith was stronger like Abraham, because God still wants to show the world that God is and that Christianity is the right way to get right with Him. But we also have to realize that everyone has a day that they will die, and that heaven is a far better place than even the cushiest life on earth could ever be.
If we have financial problems, we also have to realize that if we have dug the hole ourselves, or been overwhelmed by medical debt or a lawsuit or other devastating result, God isn't a genie that can fix the problems we created ourselves or were justly or unjustly forced into. But I also believe that God blesses obedience. Although God may not choose to fix our mistakes, have we done our part in tithing to support His work and giving free will offerings to spread His work even further. Are we giving to missions to support spreading Christianity to those who don't know the truth, for example? Do we give to benevolence to help the needy? To be clear, just doing things like this won't guarantee God will pour out blessings on you. But walking in His will in these things will guarantee that God is not actively against you (Malachi 3:8-12).
I understand that sometimes it is hard to have faith. I think that one of the disadvantages of living in the 21st century is that when the doctors say you heart is about to wear out and needs replaced or that you have cancer to use a couple of examples that are extremely high on the worry scale,. it becomes hard to have faith because those doctors will tell you that your chances of being alive x months from now are y. And frequently y isn't a very high number. And results like this make faith hard.
Do we trust in God and have faith in His promises or do we trust the doctors? And we also need to ask God if we are going through this for part of His purpose and He wants us to go through the medical stuff because His plan is to somehow use the doctor's treatment to reach someone for Him that otherwise would never know Him. That's a scary thought, but can be part of His plan as well. But if we think His plan is to heal us, then we need to have the faith of Abraham and trust God. God could well end up giving us a promised outcome that will bless the world with our testimony.
Back to the last bit of Romans 4, the chapter closes with Paul's reminder that we need to remember that the only important part of this discussion is that we need to believe in Christ's sacrifice for the remission of our sins. He is the one who justifies us to God. There is no other source of forgiveness for our offenses.