I'm planning to try asking him about that tomorrow.
Literal
Tomorrow that thing [about-について] him [to-に] ask-try intend [polite-です].
~てみる ('try doing') is added to する — 'try asking.' The nuance softens from 'I will ask' to 'I'll give it a try and ask' — lower certainty, more tentative. Japanese loves layering these modal tints.