There are many options for headphone amplifiers, from dedicated chips to opamps to discrete transistors. All are capable of performing admirably. One question was unaddressed:
Do I need some impedance matching device?
No, impedance matching is generally undesirable in audio applications. The major exception is the analog phone system. In general you want the output impedance much less than the input impedance it is driving. So your amplifier should have an input impedance much greater than 2.2 kohm and an output impedance of much less than 150 ohm. This is not hard to accomplish, just something you need to be aware of.
Is this a line-out from some existing equipment, or are you implementing the codec yourself? If the latter, you will need some anti-aliasing filter. The data sheet for the CODEC will help you out.