The following is an outline, based on some extracts from the consultancy info I sent to a colleague teaching Classics overseas regarding Carthaginian chariots:
Carthage, as you may or may not know relied primarily on her navy, but also maintained a large army made up of Carthaginian forces, and "mercenaries" although we should remember that some mercenary units were mercenaries as we would apply the term today and others were more like levys of allied troops. These included large numbers of Libyan forces, Numidians, Iberians, Greeks, Sardinians and Celts (Gauls, Celto-Iberians and Galatians).
The Carthaginians themselves used chariots in civil processions and chariot racing throughout Carthage's history. In warfare they originally seem to have used a thick and sturdy quadriga (drawn by four-horse team like Seleucid chariots.) They are first mentioned at the Battle of Crimisus River in 341 BC. They were important in the early Carthaginian army and it seems that up until during the Second Punic war the Libyans supplied the bulk of these. (Warry, John. Warfare in the Classical World. pp. 98-99.) This sounds at first a little surprising, given the semi-arid climate of North Africa, but chariots (and horses perhaps a little earlier, but not very common common until they were introduced to Egypt c. 1500 BC) appear in rock art in a number of sites throughout North Africa and the Northern edges of the Sahara - it seems that North Africans, Libyans among them had probably adopted and maintained chariotry and horse breeding for centuries prior to the rise of Carthage, although later Hellenistic and Persian influences would have been established by the Punic wars. The Carthaginian use of chariotry in warfare declined during the Second Punic war and was gradually phased out and completely replaced with war elephants.
How would the Carthaginian chariots have been used? They were drawn up in units at the front of the Carthaginian army (Plut. Tim. 25.1,27.2), like war elephants we should not be too quick to underestimate just how much of a pyschological impact this would have had on unnerving enemy troops, The chariots would then charge forward, firing arrows to thin enemy ranks, and perhaps javelins too at closer quarters, while the sounds of the galloping hooves and thundering wheels, not to mention the dust would be quite intimidating. Horses will not charge a densely packed phalanx of spears, so the chariots would then either whirl away and repeat this several times before the infantry went in, or if they had created a weak point of injured or fleeing troops in the enemy line, they could exploit this and charge through/create more havoc on this gap in the lines and then the infantry could stream through or even encircle and cut down the enemy in the wake of the chariots.
Even though the Carthaginians gradually abandoned chariotry, throughout the Punic Wars Carthaginian forces had smaller but still significant use of mercenary (allied) Celtic chariots. The Gauls were extremely skilled with their light, mobile and manoeuvrable biga (two horse drawn) chariots as Julius Caesar noted in the Gallic wars. In fact our word chariot (and car, and carriage) are derived from Latin Carrum which is a loan word from Gaulish Karros. The Celts tended to use their chariots as "battle-taxis" hurling several heavier javelins at range, but primarily to move elite heroic warriors to, from and to different points on the battlefield, although the warriors tended to dismount and fight on foot and then be driven away to their next destination on the battlefield. The exception seems to have been the Galatians that settled in Asia Minor, as they seem to have adopted the Persian scythed chariot, with rotating blades coming from the axle.
*Now, I did hear once that some Galatian scythed chariots were used by Hannibal's forces to deceive effect on several occasions early into his Italian campaign, however I haven't been able to verify (nor completely rule out) that yet.