Yes! I have a scheme, and my dp is exactly like your dh sounds. I was getting so fed up of constantly doing everything that needed doing as I spotted it, while he arsed around on his computer. He also is lovely and doesn't mind doing stuff, but he just can't SEE what needs doing or have the efficiency for example to bring dirty cups back through to the kitchen when he is going that way. Grrr.
Firstly allocate some jobs that are his jobs and he always has to do them, every day, ideally at the same time every day, then he doesn't have to remember. E.g. empty and refill dishwasher, or you could have once a week jobs, e.g. he has to hoover every saturday morning. To make it fair, you also have some stuff you do regularly. in our house dp does dishwasher, I wash clothes.
Secondly, every night after DS goes to bed (7.30), we have to do whatever needs doing for the next hour. This includes cooking the tea. So we talk about what needs doing - essentially this is me telling him, but in theory he can suggest things too - could be emptying bins, replacing lightbulbs, changing bed, whatever, plus of course the tea. Then we share out the jobs and do them. If someone is out for the evening, then the other doesn't have to do any housework. Also we don't do any at weekends apart from the daily tasks.
He likes it because he never has to think or have any initiative. I like it because I'm not doing housework non-stop because I know there will be a time when it gets done, it won't pile up, he will do his share and I don't have to nag.
Has currently worked for about 3 months!