Belfast has a temperate maritime climate: rarely very cold, rarely very warm, and wet throughout the year. The best months to visit are May to September, when the days are long, the weather is at its most reliable, and the outdoor bar terraces are in use.
Summer is festival season: the Belfast International Arts Festival in October/November is the largest arts festival in Ireland; Belsonic is a summer concert series in Ormeau Park; and the Cathedral Quarter Arts Festival in May is one of the better small city arts events in the UK. The 12th of July (the marching season's main day) is significant — the city is busy, some areas are tense, and it's worth understanding what you're witnessing if you're in the city that day.
Winter in Belfast is manageable — there's little snow, plenty of indoor life, and the Christmas market in the City Hall grounds is one of the better ones in Ireland.