What software license allows me to have GPL for academic use and paid use by industry for my open-source project? I have spend many years during my PhD developing a code base and decided to open-source the code when I finished. It's currently under GPL. The code is used by many different research group around the world, and that was my goal; to provide a useful tool for the research community.

Recently I have been contacted by a company that wants to use the software. My intention was always to charge a fee for industrial use of the code. On the website, it says something to the effect of "GPL for academic use", "please contact me for industrial licensing".

Firstly, do you think this is reasonable approach? Personally, I don't think that industry should get a free ride from my publicly funded research code.

Secondly, what are reasonable terms, pricing and software license for an "industrial license"?
Your license is not the GPL if you add additional restrictions, and your restrictions may turn out to be incompatible with the GPL in subtle ways. Consult an IP attorney.

Fortunately for you, even if your code is GPL, many companies might not want to use it because they might worry about it infecting their product. So, you might be able to charge them for a proprietary licensed version. That being said, if your work came in under a US grant from the NSF or similar agency, its terms may require you to open source the work period. Additionally, your university is likely to have an Office of Technology Commercialization or similar body that is responsible for software licensing to industry. At mine, the Board of Regents retains the rights to all software and commercially licenses it with the "inventor" getting half and the university getting half, typically, unless it is open sourced.  

You should check with your university and the grant terms that funded your work. Corporations are typically tax payers in their local jurisdictions, so it's not necessarily wrong for them to benefit much as another researcher would from your work (i.e., for free).