Paul’s question at the beginning of chapter six is a variation of a phrase used in 3:8 – “Let us do evil that good may result.”
It’s hard to understand how anyone, conversant with Christian theology, could get the idea that good comes from doing bad – but there it was in the Roman church. If grace covers my sin and makes me “right” with God (righteous), then the more I sin, the more grace I get, the more righteous I become!
In chapter six, Paul says “no no no no no!” Such a convoluted reasoning is totally contrary to what we become in Christ. We enter Christ’s death to participate in his resurrection. If Christ died to free us from sin, entering his death cannot allow us to continue in the same sin that Christ died to free us from (verses 7 and following). Second, freed from sin, we must become the servants of God. You cannot be both the servant of God and the slave of sin.
It is important just here to make a distinction: Paul is not saying a Christian cannot possibly be a servant of sin – for he very well can! Paul’s previous chapters have affirmed that these Christians were indeed sinning. Paul is saying that sin may not be a viable option for a Christian’s life – not if he intends to live in harmony with His new status in Christ.
I’ve never known a Christian to make the argument evidently the Romans were making – the one about getting more grace by sinning more. But I have known Christians to live as if they believed it, whether they actually said it or not.