Jerusalem, like Israel in general, is a year-round destination, but the very best months are late March through April, and October. Winter is colder than in Tel Aviv, but gloomy winter days are often followed by gorgeous sunny ones. Jerusalem gets its own back in the hot, rainless summer months; the inland capital is dry and cools off toward evening.
Avoid the main Jewish holidays, when hotels charge peak prices and many sights are on a different schedule. Many sights have shorter visiting hours (or close) on Fridays, Saturdays, or Sundays because of the sabbaths of the communities that operate them.