You are wise to be aware that lack of drainage is harmful to most plants in containers. You have several options. If you don't want to make holes in the washtub, you can plant a slightly smaller container with your flowers and insert the pot inside the washtub, first putting a brick or a layer of gravel in the bottom of the washtub to stand the container on.
Otherwise you can put a couple of inches of gravel at the bottom of the washtub with a layer of screening or landscaping cloth on top. This stops the soil from going through to the gravel. Then add your soil, and plant as you usually would. A third option is to make a small water garden in the washtub as it is. These can be charming and interesting on a daily basis. You can find basic directions in any water garden book at your local library or on the Web.