Which elements of chess are best to focus on for extended "crash course" learning? I want to spend between 1-2 hours/week practicing chess. I know that my skill will be severely limited by the time constraint. However, within this time constraint, where would my time be best spent? I obviously won't expect to challenge serious players and win (or even provide much of a challenge), so you can assume that all of my opponents are casual players as well. 

For example, would my time be best spent playing AI's (so I don't have to wait for my opponent to make a move)? I'm not expecting an exact breakdown of what I should be doing, just some general advice.
Depends a bit on your level. 

Assuming you are rather a beginner player, the best way to improve is by studying tactics/solving tactics puzzles. You can do this on many websites, which is quicker than doing it from a book and also most websites will present you puzzles suited to your level. 

For a beginner I find chesstempo.com quite useful because after you solve a puzzle (or after you fail to solve), it will tell you the name of the tactical motif. Take note of these motifs, understand them and try to look for them when solving puzzles. Once you have stored a number of motifs in your brain, solving tactics puzzles will be much easier since you will not have to calculate move by move, but can think in patterns. Also chesstempo have a list of tactical motifs which you might want to consult.

As a second major topic you should learn general principals of play (e.g. develop pieces to active squares, occupy the center, king safety...) and perhaps a bit later positional motifs.

Opening theory is not very relevant for beginners, because in general it can only gives a small positional edge and most beginner games are decided by blundering a piece somewhere. Also for making use of openings you need to play opponents who also play theory, which is usually not the case with beginners. So don't waste your time with openings at this point in time.

And of course you should keep on playing games. Personally I'd rather play humans (online or over the board) than computers, but whatever suits you. Play games with longer time controls (10 min or more) in order to reduce the chance for blunders and to give you the opportunity to think about the position. For me lichess works best and has lots of features, but there are other options for playing online.

How to split time between studying and playing is up to you. After all you want to enjoy playing chess.