What are the basics of making a rechargeable battery pack?

You can't just connect an AC adapter to a bunch of rechargeable batteries directly. Instead you need to have some sort of supervisory circuit to shut off (or taper) the charge when the batteries are fully charged. These days that is usually done with a specialized IC doing most of the work.

Here is a charging circuit that supposedly works with up to 8 AA NiCad cells.

The pack will have to somehow be modified to charge the cells in parallel (right?) while powering the device in series.

Yep. A relay is an easy way to do this.

See http://www.batteryuniversity.com/ for the specific type of battery you're using.

To recharge lead-acid batteries, for instance, you start out with a trickle charge to make sure the battery doesn't have any shorted cells, etc.

When it's up to a certain threshold, you do a constant-current charge. The maximum current is specified by the battery manufacturer. (You measure the current with a small current-sensing resistor in series with the battery. A 0.1 ohm resistor with 100 mV across it has a current of 1 A going through it, for instance.)

After the constant current stage, you apply a large constant voltage, and then when that stops taking current, you drop back down to a float charge.

Here are instructions for nickel batteries.

Batteries can be very temperamental, depending on chemistry, which you don't mention. If you get the voltages wrong even by a little bit it can mean the difference between long life and short life.

Lithium-ion is really several different types of chemistries but all are rather fickle for charging.

This isn't perhaps the best answer but if you really depend on having your batteries last longer, don't do it yourself -- use the manufacturer's charger.