There is no need for a trial. Ulfric is guilty of High Treason, the punishment is death. Any act against (whether directly or indirectly, or any nonaction which results in circumstances, directly or indirectly, against) a allegiated sovereign or by a vassal to a liege, resulting (or what a reasonable person would assume would result) in physical, emotional, mental, or magical harm or injury in said sovereign or liege. The punishment for this crime will be death. - Legal Basics.
There also appears to be no trial system within Skyrim.
In the Imperial City, legal counsel is available by persons like myself, but the provinces have no such system in place. Perhaps that will change in time. We can all hope so. - Legal Basics.
Skyrim has been part of the Third Empire since it was declared by Emperor Tiber Septim, after all that time there appears to be no courts of law, it would seem Nords are generally opposed to having them. Big men with big beards with big axes sitting around talking doesn't seem to fit in with Nordic culture that heavily surrounds violence.
In Nord terms, trials are probably for Milk-drinkers.