Instagram downsizes profile pictures to 320×320 and crops them to a circle. Blur comes from two habits: uploading a tiny image that gets upscaled, or uploading a giant one that the server resizes badly. A clean 720×720 square with light sharpening avoids both.
Keep the face or logo inside the circle. Anything in the corners of the square is cut. Simple backgrounds compress better than busy ones and look cleaner at thumbnail size in comments and DMs, where the picture is even smaller.