Is the Strawberry Perl c\bin directory in your path? If not then external DLLs cannot be located by the system and nothing that depends on them will work. In this case the file you refer to, mysql.xs.dll depends on libmysql__.dll. If that file is not in the path then the bindings will not work.
The .xs.dll file is also the compiled XS bindings, not the mysql DLL.
If you have downloaded a portable version of Strawberry perl, which from memory is what Chocolatey distributes, then you need to run the portableshell.bat script to set up the paths, or manually prepend them to your path. This script is at the top level of the Strawberry perl distribution.
(Edited a few minutes after posting to clarify reference to .xs.dll file.)