God has certainly asked us to do good in the world, and has never given us free rein to do as we please. 1 Peter 1:16 Be holy for I am holy.
Jesus tells the adultress to go and sin no more. We are told repeatedly that faith without works is dead James 2:26.
Now Jesus also came to fulfil the law, not abolish it. Matthew 5:17, so we can see how important the law of God is....
However, the wages of sin is death, Romans 6:23, and once we have sinned once, that's it, we're sinners. That's where Jesus came down, to receive that wage in our place. God isn't lying, the wage really is death, just Jesus stepped in and said to pay Him instead. Now we're free to enter God's kingdom if we just believe in Jesus and His saving work...
So, where do our works come in? Only really with our continued belief in Jesus that won't separate us from Him. Our other works won't save us, nor can they as we're sinners, although they will be a sign of someone with faith, because believing in God means believing in His word and the He wants us to do good works. If we don't, have we believed? Do we have any faith in Him at all?
It's worth nothing the verse: But if anyone does not provide for his own family, especially for his own household, he has denied the faith and is worse than an unbeliever.” —1 Timothy 5:8. Our lack of works can show that our faith is seriously weak, or not there at all. Clearly, works are a clear sign of faith in Jesus AND his teachings. Not just in what He did for us, but also what He instructed us to do.
We are, as I've mentioned, told to continue doing good works. Not doing so denies us so many blessings both here and after our earthly lives, as I'm sure we've all experienced on a day to day basis.
Bottom line, we could not save ourselves through our works as we just aren't good enough. Death us the due payment, before with sacrificed animals, now through our saviour Jesus, the sacrifices lamb of God. That's opened the door for our salvation, nothing else.
The fact remains that God wants us to do good, that's just not what saves us, we already failed at that. He will, however, rain blessings on us and those who decide to follow our lead in placing faith in Him. The works will be a constant sign that we have placed our faith in Jesus Christ, what He did and what He said.