I have purified His tagged proteins in the past, but I am a bit confused on how to purify doubly tagged proteins. I tried looking in the literature on how to do this, and I have not found anything useful so far.
Purification of GST followed by His would be a good approach. GST purification comes with less contaminant proteins compared to His. But there is always an amount of partially translated GST protein which would show up. If this is followed by a His purification that would essentially give you majority of target protein. Further purification steps like fusion tag cleavage (if any protease site present) may clean up the protein as well. If no protease site present a chaperone wash with 1mM ATP between loading and elution can help clear some contaminating chaperones (mainly important if using bacterial system). Gel filtration as last step would clear up giving highly pure protein. Gel filtration should be done based on the amount and purity of the protein as seen on SDS-PAGE. protein should be concentrated to a small volume to get a better separation during gel filtration if there are contaminating proteins.
Having said all this sometimes His followed by GST can work better. It will be obviously better to optimize the protocol for purification by growing up 250 ml of cultures and doing small scale purification and analysing by SDS PAGE as to which step first gives less contaminations.
affinity > (second affinity) > (tag cleavage plus another affinity chromatography) > ion exchange > (second ion exchange) > (third ion exchange or adsorption chromatography) > size exclusion > (desalting if necessary)
Same protocol should work. Based the purity of the preparation after His tagged purification and your requirement (based on what assays and weighing against contamination) add additional steps of polishing as suggested above.