If I have a set of data how can I construct a probability plot to see whether the exponential distribution is a good fit for the data?
You can plot an empirical distribution function and see how much it looks like the cdf of an exponential. I suggest using R, it had a build in function to do this (here is the reference)
The function is called ecdf. It takes a vector as an input and that should be your data. You can check out the documentation and in particular, 5.1 and 7 refer to how to use vectors and input data and 12.1.1 shows how to plot things.
R is not that easy at first sight but I think it is worthwhile to learn if you want to do statistics. If you use an other piece of software to plot things (or pen and paper) let me know and I can be a bit more helpful.
You could also use for example Pearson's chi square test to determine how well your data fit an exponential (which again R has built in).