There’s an interesting article by Elliott Gue at Seeking Alpha on whether the dollar drives oil prices. Skipping to the last chapter, the answer is that the exchange rate is a factor but it’s not the most important factor. 90% of the changes in the price of oil are explained by factors other than the dollar.
That raises the question what does drive oil prices? I can only offer guesses. My guess would be that short term movements in oil prices are influenced (in descending order of importance) by supply and demand, speculation, cartel pricing, and the value of the dollar.