What would cause my dishwasher to leak when it starts filling?
Bad Seal! Or cracked pipe. 

You will need to take it out completely.

Properly dismantle it  in and open space and trace/examine the source while the power is on

(this can be dangerous so do not take chances)

If it only happens on the start- maybe it the inlet pipe cracked, seal perished.

If you do not feel comfortable with dissembling it call a recommended local service man.