I was trying to start my until the it's dead, I suspect the alternator because in the dashboard is written battery charger. I jumped start it until its started. Should I check alternator? Or what because the car battery is only two months
Alternator has nothing to do with starting the car,
Provided the battery is fully charged.
the charging warning light is on until it runs then if the alternator is good the light goes off.
So I'm wondering on it's service state,
Air filter/plugs and so on?
You could have more than one fault with a car of that year . Could be coil pack or ignition leads ( whatever is firtted ) , compressions low to name a few . Without knowing more facts and some service history , hard to advise .Changing plugs is a start .