To have the prayer life of Paul...Paul wasn't casually telling the Romans that he remembered them and prayed for them. Paul did not waste words on the page to say something that wasn't true.
It's not like today when someone shares something with us and we tell them that we will pray for them. We say this to make them feel better and make ourselves look good, all the while we have no intention of carrying through with what our mouth uttered.
However, Paul was genuine, when he advised the Romans that he constantly remember them in all his prayers it was a fact. God was witness to the reality that Paul prayed and remember these Christians. He remembered them in all things, always thanking God the Father when he heard reports of their progress and faith.
Paul also tells informs us as a witness to how much he prayed for these people when he relates the genuineness of his service to the Lord, the Amplified Bible says it this way, "For God is my witness, Whom I serve with my [whole] spirit [rendering priestly and spiritual service] in [preaching] the Gospel and [telling] the good news of His Son, how incessantly I always mention you when at my prayers." He is reminding them of how he serves the Lord in everything, with his whole spirit. He served the Lord passionately and wholeheartedly.
News of Paul and what he did traveled to the known world. Paul was known for being an intense and driven follower of Christ and signs and wonders followed him by what he did. Paul is simply reminding the Romans, that in the same way I serve the Lord wholeheartedly and fully in my whole spirit, this is the same way I remember and pray for you before the Lord... He is my witness.
Today, if you have made commitments to pray for people or want to pray for people but do not know where to start... just start thanking the Lord for their faith and what He is doing in their lives. Remember them before the Lord and praise the Lord because of His goodness towards them. You will find that it is a pleasure to pray for people because you are making it all about the Lord, and this brings Him glory!