I don't know what the answer is. DD is 1yo and is exactly the same but has pretty much always been like this. We try to limit chores and stuff to when she is asleep but if we have to do something then she has to scream.
If I'm getting dressed she goes in the cot otherwise she either flings herself off the bed towards me (and misses hitting the floor) or she's on the floor pulling and clinging onto my leg and I can't get anything done.
Downstairs we have 1 safe room or the highchair to place her in.
Sorry no real gews of wisdom except that this is a classic seperation anxiety age, it is a PHASE and will pass - make that your mantra!!!
Set him up with an activity and play with him once he gets absorbed try leaving him for a few moments and then go back to join is with him again. Is he napping much in the day, just thinking that will give you a break!
I deal by going to work and leaving dh with the little darling She is worse with me though and screams when I leave for work.
Mostly it doesn't bother me because I am naturally lazy. There are only 2 things that really bother me are when she constantly gets up and down off my lap (straight up and down with no pausing in between for ages) or when ds sits on someones lap she has to climb up as well and try to push him out - that pisses me off royally.
Ds was a very screamy baby though so I can kind of switch off.
He's probably working on the 55 week developmental spurt - in the run up to developmental leaps, children become 'clingy, cranky & crying'. You can read about it in The Wonder Weeks. It's usually worst with the mother (or whoever is their primary carer).
It should ease off once the leap is over. Researchers describe the effect on babies as being as if they've woken up and discovered themselves on a different planet - they need lots of reassurance and help till things settle down again.
I find a sling on my back helps when I really have to do some housework. Even if DS isn't too impressed, at least I know he's safe & with me.