How does “Reputation” work?
I'm banned from asking, and my programming skills are weak compared to the average stackoverflow user. What would be the best way to gain reputation in my current state?
There are several tags such as jquery and php where chances of easy-to-answer questions are pretty high. So if you have some knowledge in these areas add those tags to your favorites and look for questions you can answer.
I think you should focus on lifting the ban and not on "what else" can get you reputation...
The ban will be lifted as soon as the system detects you are actively contributing to the site. That is the thing you should be focused on. That'll let you get back to asking questions. Looking at the posts you currently have, I think there is an underlying problem of knowing how to ask and exactly what to ask on Stack Overflow itself.
The last item might be a good option for you as well. There are other Stack Exchange sites dealing with different questions and topics like -
Perhaps there is another site that catches your eye and once you reach 200 rep on at least one site, you can associate your Stack Exchange accounts with each other. You'll receive a 100 point association bonus on all other sites!. This bonus is to get you past the initial reputation barriers and allows you to contribute more actively than a brand new user with only 1 rep point.
As @ThiefMaster pointed out, there are some very active tags that contain many simple questions. Why not try answering some of those? I see that your current posts are related to the android tag; I'm sure there are some simple questions there that you could answer...A couple of upvotes will get you back on track! The high traffic tags get loads of attention so all you have to give is a correct answer, throw in some formatting, and you are bound to get some rep plus the ability to ask questions again...
Another thing that would get you some reputation points as well as assist the community, is to suggest some edits to existing posts (that are not yours). Each approved edit will give you 2 rep points. You can carry on like that until you gain the edit privilege (2K), once you have that, you will no longer receive reputation for accepted edits - because you can now make edits without having to get them approved! This suggestion, however, comes with a warning. Do not make small trivial edits just to gain reputation. If you see a post in need of some editing, make sure you take care of the whole post. Check the title, capitalization, grammar, tags and anything else that doesn't contribute to the post. If you are not 100% sure then it's best to leave it and let someone else handle it. If the only thing you are editing is removing signatures or greetings/thanks and other than that the post is great, leave it. No need to clog up the edit history with minor edits... To many edits from too many users have consequences.