Capital punishment doesn’t seem to be an issue of dina d’malchus. Paying taxes, property ownership, legal contracts etc…are what most gemaras and rishonim discuss when dealing with this topic. Perhaps the rambam would include it as he seems to view dina d’malchus as an offshoot of dinei melech which could incorporate anything the king wants to do.
Killing people in response to committing crimes is a very interesting halachic issue but is probably more of a question with regards to dinei nefesh than dina d’malchusa dina.