It's a very broad question with no single answer, of course. But the dates of your question give us some meaningful context, in that in 1787, the Constitutional Convention began, and in 1857, the Supreme Court ruled in Dred Scott vs. Sandford that slaves were chattal property.
So we can generalize that the policy of the US government, as well as its intent, was to preserve the institution of slavery during these years, or at least not to pursue abolition, and they were quite successful at doing so. The Constitutional Convention purposefully avoided the question of slavery, for fear no unity between the states could be reached if a decision to abolish or keep slavery permanently was pursued. This is why the Constitution before the 13th amendment says virtually nothing about slavery. The other major motivation of the US government in this time period was to keep the country from splitting apart over slavery and its expansion, and they were somewhat effective in doing so by using political compromises that allowed the expansion of slavery west, but with limits.
The Supreme Court, throughout this entire period, upheld the property rights of slaveholders and refused to recognize slaves as human beings.
The Compromise of 1850 is where it started to unravel, in that popular sovereignty in the Kansas Territory was a disaster, and led to a bloody prelude to the Civil War.
There are two ways to look at how effective the government was in dealing with slavery. You can say that the government dealt with it effectively because it managed to put the issue off and keep the country together for more than 70 years. On the other hand, you can say that the government dealt very ineffectively with the issue because the issue helped lead to a devastatingly horrible civil war. Which you pick is up to you.
The US government's major intent with regard to slavery was to keep the country together. It wanted to allow both the North and the South to coexist in one country even though they had such different economic and social systems. Because of this, the government tended to try to create compromises. These compromises (like the Missouri Compromise, the Compromise of 1850, and the Kansas-Nebraska Act) tried to give each side something to keep them happy. This approach worked for 70+ years, but then it fell apart when the differences got to be too wide to paper over with compromises.