The wikipedia article on impeachment defines the process as a trial, analogous to charges brought by a grand jury, and for which the legislative branch considers evidence. That sounds like due process to me.
It is not. Congress is not required to hold any sort of trial if it does not wish to. If it felt like it the House could vote this afternoon to impeach Kavanaugh without holding a single hearing or providing any evidence and the Senate could convict him ten minutes later, also without holding a single hearing or providing any evidence.
If you were to be imprisoned as the result of a process where that happened your conviction would be overturned as a violation of your Constitutional right to due process. As no one has any due process rights in regards to federal employment and Congress retains the exclusive authority over impeachment 1) the right to it doesn't exist and 2) requiring it would be unconstitutional.