This is a common complaint of parents of ADHD children.
First, I would make sure that his ADHD is appropriately treated. If his doctor, or whoever diagnosed him with ADHD had recommendations for treatment (typically, that includes medication), then you'll need to make sure his treatment is optimal so that he can do the things he needs to do.
You mentioned that he hears well and can even concentrate well when he's read to. The key to understanding ADHD is that children (and adults) often can concentrate quite well- even too well (we call that hyper-focusing) when the activity is intrinsically of interest to them. If your son loves to be read to, he'll have an easier time paying attention.
That's why we see so many problems in school, in classes where the child might have little or no interest. In those cases, he/she often has great difficulty connecting, paying attention and retaining the information.
It's also important to understand that many, if not most children with ADHD, seem to develop a bit later socially than their peers. At six, your son may be developmentally a bit younger.
Putting this all into perspective for your son, it could well be that what he's being asked to remember are not things that are of interest to him. Or perhaps, and more likely, he hears you but simply forgets. That's the nature of the "beast" in ADHD.
To help him, you'll need patience and creative thinking. Many with ADHD do quite well with visual cues. Consider writing up your "to-do" list for him and post it in several places where he is likely to see it. Number the items, add interest with stickers or whatever, and when you ask him to do something, reference it also by having him look at the list.
Also, try asking him to repeat back to you what you requested of him. Using more than one modality (hearing, seeing, etc) often works best. You want to make sure you have his attention when you ask things of him, so make sure you are getting eye contact from him.
Never give too many demands at one time. Many with ADHD have a hard time processing more than one or two things at one time. So you may want to do something like this:
"Tommy, we need to work on cleaning up your room. Can you spend 10 minutes just picking up your clothes and throwing them in the bin?"
Then set a timer so that he has a sense of how long he needs to work on this task. Once he's done, go to the next task.
In other words, break each chore/task down into smaller parts.
We tend to talk too much to our ADHD kids about what they need to do. Usually, short requests, using visual cues, and keeping them on routines and giving them structure, is the way to go.
Hope this helps.