"President Donald Trump Goes After Own Party At Press Conference"
The "his" before "own party" has only been dropped because it is a headline or title. Author is being lazy or more likely it's just done to save words so the title will fit easier in the allocated space. The 'his' here is implied. This 'cutting' of words is common in newspaper headlines, title of newspaper stories, or in this instance the title of the video. If you were using this same sentence in the body of an article (not the title) then the "his" should be included.