Summer is the obvious time to visit — Läckö Castle is open from late May through August, and the Läckö Slottsopera opera festival runs through July and August with performances in the castle courtyard. Lake Vänern is cold but swimmable, and the long Swedish summer evenings make the lakeside setting particularly pleasant.
Spring is worth considering if wildflowers are the draw. The Kinnekulle plateau produces its best display from late April through May: pasque flowers first, then cowslips, then several orchid species through June. Crowds are thin and accommodation is easy to find.
July and August are the busiest months — the opera festival draws audiences from across Sweden, and Läckö gets genuinely crowded on summer weekends. Book ahead if you're visiting then. Outside midsummer the town is quiet year-round.
Winter offers little beyond the pleasure of a small Swedish town in the dark — the castle is closed and the countryside is cold. Stick to the warmer half of the year.