Is it proper to use this instance of two consecutive adverbs separated by a comma?
e.g. "He seriously, actually has this problem."
I have seen some consecutive adverbs be okay, but others not. Thank you in advance!
Sure. The comma stands in for an elided conjunction. From The Lodger by Marie Belloc Lowndes:
Again she glanced at him quickly, furtively, but he seemed just as usual
Your example is a little odd because while people have serious problems, we usually don't say that they have them seriously, i.e., in a serious way. The sentence also has an echo of an introductory adverb --
Seriously, he actually has this problem
which means, "I'm serious about his having this problem."