Place in a blender and blend on high for 1 minute. For a smooth puree, I found that my blender worked better than my food processor.
If you want to remove the seeds, pour your strawberry puree into a fine-mesh sieve over a bowl and stir the puree until all that remains are the seeds. My experience is that the seeds and related pulp amount to less than 1/4 cup while the puree will be between 1 1/2 cups to 2 cups.
Before you add sugar, taste your puree to make sure it needs it. If you do add it, do it in teaspoonful increments until you've reached the desired sweetness.
Refrigerate for up to 2 days or freeze for up to 6 months.