I don't have a Missouri specific answer regarding appeals, but here are several starting points for gathering information for your paper.
The Death Penalty Information Center has a great many resources to aid in research, including state-by-state information: see this site
Jim Voelker, Deputy Circuit Librarian, St. Louis, has posted a Death Penalty Research Guide at this site.
The Alabama Attorney General's office has put up an extensive Guidebook to Alabama's Death Penalty Appeals Process on its website.
Here are discussions of the appeals processes in two more states: Oregon and Texas. You can get an idea of the process from those. In both of these cases there is an automatic appeal to the State Supreme Court for capital cases, while in Alabama the automatic appeal is to the Court of Criminal Appeals.
The general appeal process is summarized at the bottom of this page by the ACLU.
Depending on the depth of your research needs, you may want to check out the National Archive of Criminal Justice Data – you can use a guest log-in to get at some of the data. See for example this page.
Missourians to Abolish the Death Penalty (bias alert!) has this page of links on its site: links.
Finally , there's this handy little diagram about the appeals process on this page from the McClatchy Washington Bureau.
