Helsinki is undoubtedly quite cold almost throughout the year so be warned and carry at least some winterwear even while travelling in Summer especially if you are coming from a warm country.
Winter: Novemeber to March is the long and lengthy frosty winter that the city sees. It might be daunting for some but there are a lot of opportunities not just to see snow but try out ice sports during this time. It isn't the peak season and hence prices are low. If snow sports are on your mind you can go in winter. Several festivals also take place during this time.
Fringe months: April and May see winter slowly receeding. The weather is cold but it is a good time to visit since it isn't frosty. You will also be able to get great deals during this time.
Summer: Summer begins in June and ends in August; which means it is a really short season. Summer isn't sunny but is a great time to try out all the outdoor activities. The weather is very mild and pleasant and hence very popular wih tourists. Travellers certainly make the most of it.
Autumn: September and October are the months when the temperatures begin to decend. While it can get quite chilly, It is a good time to visit for budget travellers.