People talk about a weak economy and weak currency almost like they are the same thing. Certain commentators like Peter Schiff encourage this view. However, the correlation in the US has actually been the reverse.
When there is growth, a lot of things are pushing down the dollar at the same time the market goes up.
1. Increased demand for commodities weakens the US's terms of trade.
2. Re-leveraging allows people to increase their bets against the dollar.
3. Many other countries (excluding Japan) have interest rates that react more strongly to global growth, so an economic recovery means that the interest rate differential will become less favorable for the US.
4. When the Federal Reserve promises weaker monetary policy than expected the market reacts positively.
Of the variables listed here, numbers one through three will most likely reverse in a downturn not centered around a dollar crisis. People in macro often focus on reason number 2, calling any dollar strengthening during a downturn a "Dollar carry unwind" in reference to the people who have been short the dollar and long a higher yielding currency that have to close out their positions during volatile markets. This explanation is probably underweighted by people following the markets from day to day, but correlation it describes is still accurate even if the causal reasons behind it are not.