For a particular stock, what's the simplest way to calculate betas for the Fama-French factors SMB and HML?
Not sure what the question is. As John points out: the method is linear regression.
For the data you could look at Kenneth French's wegpage for US stocks. In the wikipedia article you find the links to factors for other countries (UK, Germny, Switzerland) - though I have not checked these links.
Note however that the Fama-French model works better for portfolios than for individual stocks.
First of all, the only one way to compute factor betas is to use the linear regression model, as suggested by John in the 1st comment. There is not other way to get them. You can get it by simply using excel through the Data Analysis package or using the relative command/code in other statistical command; in Stata, for instance, the command regress gives as output all statistics you need for(betas, st. dev., R^2,...).
Secondly, the Fama-French model is portfolio-based. So, following their model you can compute beta factors on stock portfolios, built ranking stocks on their characteristics (the most used is the size or mkt cap); it would have been no sense to compute betas for each stock in the market you're analyzing, since you did not conclude anything.
Indeed, all regression-based model in financial economics, as the one you cited in the question, are portfolio-based and not stock-based.
Anyway, you can compute betas for each stock theoretically, but you'll not get economically explicable output.