If you want to really emulate the casino experience, you could have the two hands side-by-side, and start dealing cards on top of each other with the left side showing(see 1st picture). Then you could use the counters above with some indication as to which hand you are playing.
The logical motion would still be left to right, i.e. you get T K, and split, the K goes to the left side of the screen, while the 10 goes to the middle. Then you work on the left hand until it is stood, busted, or doubled, before moving to the hand on the right side....that is until you get into multiple splits.
Now that I think about that, it might be better to have a preview of your "yet to be played" hands off to the right (see 2nd picture), while the primary hand stays with the current gameplay. It gets complicated with multiple splits, but I think this way you could still show the hand totals for each hand and it's clear which hand you are playing. If another 8 got dealt when you were on hand 4, and you split again, that new one would become hand 5, etc.
You'll have to figure out when the 2nd card gets dealt on each hand, I've seen some people deal it immediately after splitting and then giving you a chance to hit on the 1st hand, while others split and let you finish one hand before giving a card to the next hand.
...then there's splitting aces: many casinos only allow one card on split aces(which sucks!), this might need to be a setting in the prefs.
...some casinos only allow one split, which would help keep the interface clean, but also sucks.