I did a podcast for cfr.org that presents my thinking on this topic.
The simplest reason why oil is up and the dollar is down is that the world economy has been far stronger than the US economy. Weakness in the US economy translates into a weak dollar. Still solid global growth translates into strong demand for oil at time when supplies are a bit tight.
It is also striking, at least to me, that the countries that subsidize oil consumption the most also tend to peg to the dollar or manage their currencies against the dollar. US economic weakness consequently has translated into low US interest rates -- and low US rates have translated into low nominal rates - and even lower real rates -- in the other, booming dollar zone economies. See Martin Wolf. Combine low real rates with subsidized (or at least below-world-market-prices) oil and there has been a big increase in demand for oil in many countries that peg to the dollar or manage their currencies against the dollar.
I also was persuaded by the analysis of Goldman’s fx team. They argue that there are fundamental reasons to think that a rise in the price of oil should be bad for the dollar. The US economy is energy and oil intensive. The US has the largest existing external deficit of any major oil-importing region. The US exports relatively little to the oil-exporting economies. And the oil-exporting economies seem a bit less inclined to hold dollar-denominated financial assets than in the past.
That said, I wish I had concluded by noting that there are two clear paths that could end the current "oil up, dollar down" pattern.
Weakness in the US economy could drag down global oil demand, pulling both the dollar and oil down. Asia’s 1997-98 crisis led both Asian currencies and the price of oil to fall.
Or a rebound in the US economy could push up the dollar while adding to oil demand. In 2000, a booming US pushed up oil prices and the dollar.
The dollar isn’t always weak when oil is strong. And the dollar isn’t always strong when oil is weak. But so long as global growth is far stronger than US growth, there is reason to think that oil prices will respond to global demand while the dollar will reflect conditions in the US.