Which distribution to use with MCMC and empirical data? In class, we've been learning a myriad of really interesting techniques to sample from a given distribution, filter online data, particle filters, etc.  
My issue is that when I take some real-world data and plot it, the distribution is clearly not Gaussian.  So, I need to estimate some distribution.  Or, in the case of an online filter (particle, etc.) I need to estimate some form of transition kernel.
How do people normally do this?  What would be considered "best practices" for developing some distribution to fit empirical data?  What are some reliable "goodness of fit" tests?
 A: Kolmogorov Smirnoff is always a good test to see if an arbitrary distribution fits. You can use the test cited below to see if two sets of data came from the same distribution: 

Li, Q. and E. Maasoumi and J.S. Racine
  (2009), “A Nonparametric Test for
  Equality of Distributions with Mixed
  Categorical and Continuous Data,”
  Journal of Econometrics, 148, pp
  186-200

This test is available in the np package in R as the npdeneqtest() function.
Choosing a good distribution is always difficult; what does your data look like? Gamma distributions are rather flexible for positive data, most data can be reasonably approximated with mixtures of Gaussians, the Beta distribution is extremely flexible for data between zero and one. 
A: Note that goodness of fit tests can only rule out distributions, they don't prove which distribution the data came from.  And in many cases they may have low power to rule out some distributions, so you really don't know if the data comes from that distribution, or you just don't have the power.
But note that you can have a population that follows a normal distribution exactly (or at least close enough), but data sampled randomly from that distribution does not look as nicely bell shaped (or any other distribution).  The population distribution is more important that the sample distribution.  One thing to try is to plot several samples and see how different they are, then see if your data fits into that variation scheme.  This idea is detailed in: 

Buja, A., Cook, D. Hofmann, H., Lawrence, M. Lee, E.-K., Swayne,
       D.F and Wickham, H. (2009) Statistical Inference for exploratory
       data analysis and model diagnostics Phil. Trans. R. Soc. A 2009
       367, 4361-4383 doi: 10.1098/rsta.2009.0120

If you still feel the need to find a transformation to get to normality, then consider using the Box-Cox transformations.  The boxcox function in the MASS package for R will find an optimal transform, but it also gives a confidence interval so that you can bring outside knowledge into the decision, for example the "best" value of lambda may be 0.4, but if a square root transform has scientific merit and 0.5 is in the confidence interval, then that is probably more reasonable than going with the 0.4.
A lot of this also depends on what you plan to do with your data or the tranform of it.  Often we can apply the Central Limit Theorem and the distribution of the population then does not matter (as long as we believe that it is not overly skewed or has extreem outliers).  Or there are non-parametric methods that don't rely on assumptions about the population distribution.  So the best approach depends on what you plan to do with this data.
A: There is no definitive answer to your second question, since all the method in statistics are dedicated to developing distributions to fit the empirical data. So the "best practice" would be finding the appropriate statistical model, which might have generated the data.
A: Without some extra context the question is difficult to answer. What is your real-world data? Models (a theoretical distribution for your data) come from applications, not vacuums. There isn't one best way to approximate an unknown distribution in practice. There isn't even one "best". As a general comment, you can get a long way mixing normal distributions. But without assuming something you're going to have a hard time, particularly when the data aren't iid.
