95 Elantra-lights won't come on

Give the year and that it is now 12 years old, there is a possibility that corrosion of connections is a problem. Especially for grounding. Check the fuses, pull them and reseat them. The check and possibly replace the relays for the headlights. I doubt the battery is at fault or the alternator but the grounding points of the car and there can be several may be corroded and need to be removed, cleaned and reseated.