2002 Elantra alternator went out on car. Changed the battery and the alternator myself. Now the car starts but spits sputters and dies after and minute or two.

Have the charging system checked for proper operation. If battery is not being charged properly and goes weak the electronic control modules within the vehicle will start shutting down and cause all sorts of drivability issues

check you have good earthing points, also check the earthing strap from engine to shell. main earthing point follow the negative lead it will go to the shell.

You can't do any testing, until you have a battery
that will take and hold a charge

Then you move on to testing the Alternator,then
Fuel Pressure & continue on with other testing

If your battery light indicator went on it indicates that the battery is not charging because of either a dying battery or alternator or both!
With the red battery indicator on, the battery will eventually drain out (especially if the alternator is faulty) to the point where it will not provide the required power to the car electrical system ... thus causing it to 'sputter' and eventually die out - a sure sign it that you cannot restart the motor due to a weak of flat battery!
Hope it helps you understand the possible problem!

Try disconnecting the alternator for a few minutes.
It can run on just the battery that long.
It is possible the new alternator is producing too many spikes that disrupt the car's electronics.

crank sensor will cause similar stalling issues. You may have got a bad alternator, if the battery light is on after you jump it, probably needs a new, new alternator. make sure all your connections are tight and plugged in, both on battery and alternator.

best to put it on the diagnostics, it could be the auto chock that kicks in it just thinks the engine is cold,

It could really be any of those things. Those are normal wear and tear items that would be changed in a tune-up, when's the last time you had one?

Major vacuum leak? Do vacuum test. Clean
injectors and check fuel filter. Check timing
too.

Does the battery light turns off when you start the car? You can check the main fusible link by the side of the battery.(60A or 100A) this could be the problem