He said, "I have not come to abolish (or do away) with the law, I came to fulfill it." If "fulfill" means "do away with" then that verse is contradicting itself, which we know it can't be. The word for fulfill has an interesting entry in Thayer lexicon, basically it says that it means to fulfill in the sense of making God's promises known and the manner in which He desired the law to be walked out to be made manifest. In other words, He didn't come to do away with it, He came to show us how God expects us to walk it. The next two verse after that (Matthew 5:18-19) support that. There are other verses too that are just as plain. Consider this one...
Matthew 23:2 saying: "The scribes and the Pharisees sit in Moses' seat.
Matthew 23:3 Therefore whatever they tell you to observe, that observe and do, but do not do according to their works; for they say, and do not do.
Moses' seat was the seat within the synagogue that the Pharisee or Scribe would sit and read the Torah in Sabbath. What Yeshua is saying is DO what they say from that seat, i.e. do the law... but don't do as they do because they are hypocrites.
Amen! Preach!
As Paul said clearly,
"For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh: That the righteousness of the law might be fulfilled in us, who walk not after the flesh, but after the Spirit." Romans 8:3-4
Therefore, those who are truly born again, and filled with the Spirit, and walk after the Spirit, will be living lives that mirror the same righteousness once revealed in the Law.
I'm not talking about ceremonial laws. Those types and shadows pointed to Christ, and Christ BECAME those things to us.
He is the only Sacrifice our sins and uncleanness will ever require.
And He is our Sabbath Rest, as we who have believed in Christ, have entered into Rest. (Hebrews 4:3)
And in cleansing the Gentiles, he broke down the wall of separation between the clean and unclean meats.
These things are fulfilled in Christ.
The glory of the old covenant is overwhelmed by the glory of Christ.
The condemnation of the Law is lifted. The curse is done away, for those in Christ.
But to those who do not believe, who are not born again, the Law still stands, and will be the standard by which the ungodly will be judged eventually. The Day of Grace suspends sentence, that all may find forgiveness at the cross... but the Day of Judgment will lower the sentence of eternal death on all who refused the only Way to fulfill the Law: Jesus Christ our Saviour.
We do not find justification by law, but by grace, through faith.
But if we are truly born again, we will be new creatures, pursuing holiness.
The same holiness of God once revealed in the moral code of the Law.
We are not saved by being obedient, but those who belong to the Lord live according to His will and not there own. So, there is room for law and grace. In fact, seeing it is the law that is the standard that declares we need grace... then it is a sound argument to say that without the law there is no need for grace!
We should WANT to be like him. We have so demonized the law that the idea of walking in his footsteps, using him as our model (he did follow the law after all and not so that we didn't have to) is just more than we can conceive with developing a migraine! 
