The dry season from May to September is the only time to visit Huaraz for trekking and climbing. During these months you can expect clear skies almost every day, stable trail conditions, and cold but tolerable nights at altitude. June and July are peak season, when the mountain huts are full and the Santa Cruz trail is busy. May and September offer the same weather with significantly fewer people.
The rainy season (October to April) brings daily afternoon downpours that can turn trails to mud and obscure mountain views for days at a time. The mountains are not inaccessible during this period, and you will have guesthouses largely to yourself, but multi-day trekking is genuinely difficult and unpleasant. The main exception is the city itself — Huaraz functions year-round and the museums and day trips to Chavín or Yungay are feasible in any weather.
Altitude affects everyone regardless of season. Spend at least two days in Huaraz at 3,050 metres before attempting anything above 4,000 metres. The standard advice — rest, hydrate, do not drink alcohol on the first night — is sound.