Carthage is a sub-tropical Mediterranean climate — hot and dry from June to September, mild and wet from December to February, with two excellent shoulder seasons.
Spring (March to May) is the best window: comfortable daytime temperatures of 18-25°C, wildflowers across the archaeological zones, occasional rain but mostly clear skies. Most of the sites are exposed and have very little shade, so cooler temperatures make a real difference.
Autumn (October to November) is similarly good — warm but not punishing, mostly clear, the sea still warm enough to swim if you go up to La Marsa or Sidi Bou Said.
Summer (June to September) is harsh. Temperatures regularly reach 35°C and the unshaded ruins become brutal by midday. If you must visit in summer, start at sunrise and finish before noon, then retreat to a café in Sidi Bou Said for the afternoon. The Tunis summer festival (concerts and theatre in the Antonine Baths) does happen in July and August but evening is the only sensible time to be out.
Winter (December to February) is mild and quiet — daytime highs around 15°C, occasional heavy rain, very few tourists. The cafés in Sidi Bou Said keep operating; most of the archaeological sites remain open with reduced hours.