OP, you don't NEED an external amp, it is recommended for better performance from your speakers. However you are going to get better quality from your new speakers and head unit then with the stock system. It's basically to be had in steps and stages, options as well. However keep in mind dropping 300+ on a set of speakers and only running them off the 18-22 watts RMS of a head unit is basically chocking their performance.

Big basically summed up everything already discussed in this thread.
However, you will not be "choking up" the performance of your speakers by only feeding them 20 watts or so, as he put it. In essence, that statement says that your sound quality is degraded every time you lower the volume. That's just not the case.
I have heard people say things like "these speakers love power", or these speakers "come alive" with more power. Its all BS. the only thing they do with more power is get louder.
An amplifier does (or should do) one thing, and one thing only. Amplify. it does not color the sound in any way, shape or form. if I does, it is of poor design.
Conversely, speakers are meant to do one thing, and one thing only...Recreate the signals they receive. Some do this better than others, and that is why you can pay $10 for one, or $1000. Linearity also comes into play. To say a speaker sounds BETTER with more power is questionable, at best. To say it is LOUDER is a more accurate statement.
Your system is only as good as its weakest link.
Buy the best stuff you are comfortable paying for.

So how does this look?

HU: avic-5000nex (chose this because I really don't need the HD radio)

Speakers: Alpine SPR-60C

Amp: Precision Power P900.4 Phantom Series Class D 900 Watt 4-Channel Amplifier

Anything you would change or anything else I need?

If your going with an amp go with something a littler higher quality. Such as one of these.

MB Quart RA400.2 Reference Series 400W 2-Channel Amplifier

JBL GTO-504EZ 4-Channel GTO Series Class A/B Car Amplifier

Alpine KTP-445A (ktp445a) Plug and Play Head Unit Power Pack for

Given a choice of the 3, I'd go with the JBL.

Soundstream REF2.370 (ref2370) 300W, 2-Channel Reference Series also worth a good hard look.

So I keep seeing some of these amps at 2-channel compared to 4-channel.

What is the tradeoff of going with 2 instead of 4?

Once again, pardon my ignorance.

You can bridge a 4 channel amp into 2 channels and produce more power from them. Or a 2 channel just gives you 2 channels you can use for your speakers.

OP, what did you end up buying?