Those are two adverbs. An adverb won't generally modify another adverb (though they can, more on which below), so they would tend to be read as both modifying the same verb, walked. If you want to use two adverbs to modify the same verb (or adjective), you need to use a comma or a coordinating conjunction. If you are saying they are both true at once, you want the conjunction that indicates logical conjunction (two related meanings of conjunction there), which would be and. If you are saying they are both true at once, and that is surprising, you could use yet.
Now, there are some adverbs that often modify other adverbs. Those are generally adverbs of degree, like 'very' or 'slightly', or characterising the other adverb in some way, like 'surprisingly'. However, there's no absolute rule restricting this to some set of adverbs. It's just a matter of being used to certain sorts of combination. No-one would look twice at "seriously quickly", for instance. Your example is meaningful, but it's jarring because it's just not used. If we take "seriously" as an adverb of degree (rather than simply characterisation), essentially making it mean "very", the meaning is clear - and people would be thrown less by that combination. If we take "seriously" to mean "in a serious manner", it's harder to see what it could mean.
If what you want to say is that he walked home whilst seriously angry, and that was apparent in his manner, I would go with "seriously angry". While that doesn't technically imply that it was apparent in his manner, people will tend to read it in that way - especially if there's some extra narration about some other behaviour that shows the anger, like kicking at rubbish bins (or trash cans in American English) as he went.
