Permitted routes for Cardiff Central to Liverpool Lime Street can be analysed at the following link:
http://trainscanbecheaper.info/permitted.aspx?from=CDF» &to=LIV&v1=&v2=&v3=&v4=&v5=&ShowRoute=Y&AnalyseMaps=YThe shortest route is always permitted and that is the one via Chester and The Wirral. You wish to travel via Birmingham so we need to look at the alternative mapped routes to see if that is allowed.
From the list of mapped routes you can see that via Birmingham is permitted on a few of the options. One being maps CE+CC.
You just need to trace a route on the maps in the order given (reverse for the return journey) without doubling back through any station/routeing point. You could go Liverpool - Crewe - Birmingham - Bristol - Cardiff for example. There are many other options via Birmingham however.
if you're not sure, then let me know exactly which stations you'd like to travel via and I can double check whether it's permitted.
Note: The direct .pdf links to the Routeing Guide maps via trainscanbecheaper.info are slightly out due to additional maps being added recently to the Routeing Guide. You may need to scroll up or down to see the correct map as referred to in the list of permitted routes.