Late May through early September is the best window for Bishkek — warm to hot days, cool clear nights, and the surrounding mountains accessible for trekking. July and August can be uncomfortably hot in the city itself (35–40°C is common) but a 40-minute drive into the gorges drops the temperature dramatically.
September and early October are perhaps the best time to visit: still warm in the day, cool in the evening, the surrounding hills turning gold, and the year's last mountain trekking still possible. Winter is grey and often polluted (the city sits in a basin and the air quality deteriorates badly under temperature inversions), but the lower trails of Ala-Archa are still beautiful in snow.
Time a visit around the World Nomad Games (held every two years, usually in early September, location varies) for spectacular horse sports and a strong injection of traditional culture into the city.