It is financial common sense that oil price and US dollar exchange rate show negative correlation. However, both oil price shock and US dollar could be leading factor in this co-movement. On the one hand, depreciation of US dollar precedes increase of oil price because oil is priced in dollar. For examples, a depreciation of US dollar could encourage global demand for oil due to increasing purchasing power of import countries. It could also lead to oil producer rising oil price to counterbalance the reduced purchasing power of the oil revenue. On the other hand, oil price increase could lead to dollar appreciation. Specifically, to the extend that oil revenues are used to purchase goods and services disproportionately from U.S, or invest disproportionately in the U.S, this recycling of petrodollars could be associate with a stronger dollar.
From research paper of What Drives the Oil-Dollar Correlation? by Christian Grisse form Federal Reserve Bank of New York, oil price shock is more likely to predict exchange rate within a month, because exchange rate respond more quickly, whereas oil price takes longer to fully adjust. In longer time horizon, say a quarter or a year, dollar exchange rate tend to predict oil price.
Not just for oil, other major commodities typically follow and inverse relationship with the value of the dollar. When the dollar strengthens against other major currencies, the prices of commodities typically drop. When the value of the dollar weakens against other major currencies, the prices of commodities generally move higher.
As you could predict, countries that are major importer of oil tend to have currency negatively correlated with oil price, whereas exporter tend to have currency positively correlated with oil prices. For example, the Canadian dollar and New Zealand dollar, along with some other currencies of nations that rely heavily on the export of raw goods, tend to strengthen with rising oil and natural-gas prices. Japanese yen tend to fall with oil price surge as Japan rely heavily on oil import.
There is a lot of theory regarding exchange rate determination. For high frequency data, exchange rate is affected by various amount of macro-economics news . Therefore when forecasting exchange rate in the short run, it is important to partial out all the aggregate of these effects before looking at the predictability of oil price on exchange rate. For longer term exchange rate determination, all the short term shock will not matter since efficient market will fully adjust exchange rate to long term equilibrium level.