Our relationship to the Law changes from demand to supply, from requirement to free gift, by our death to it in Christ.
The Law still exists, and this is why Paul warns us not to go to our old relationship with it.
We are dead to the Law through the Body of Christ that we might be married to another, that we might bear fruit to God.
Ironically, the fruit fulfills the Law by the resurrection life of Christ inside us—but only when we die to the Law as an externally binding force.
Law never changes; we die to Law; new life in us fulfills Law in stages of growth.
If we go back under Law, our spirit dies, sin revives in us and bears fruit of flesh for death.
I drew this for illustration (apologies terrible artist):
http://imgur.com/gN0SMNM
So how does Jesus fulfill the Law and not abolish it? Three ways:
1. Jesus fulfills the Law by a perfectly lived life qualifying him to be a substitute for all mankind creating a new race as the 2nd Adam.
2. Jesus fulfills the Law which demands the death of all sinners, even for one small infraction, by tasting death for every man and bearing the punishment of their sins.
3. Jesus fulfills the Law by his new life in us, when we reckon ourselves dead to the Law in him and risen by the life of God.
Some might argue that no longer being under the Law is abolishing it, but remember: what the Law demands is the death of sinners and the wrath of God for the wages of sin. By freeing us from the Law Jesus did not abolish the Law itself, he actually fulfilled every tiny jot and tittle of it perfectly: Perfect life, perfect death, perfect resurrection breaking the bonds of sin and death by a power unknown to anyone, the infinite power of God.
We become free from the Law's ministry of condemning us in Adam to find the freedom of resurrection life in Christ.