How to upload and get objects in private s3 bucket through CloudFront?

There are many ways to accomplish that, but I would expect you don't want to allow random outsiders to upload whatever data into your S3 bucket for distribution through CloudFront. You would probably want to have the picture initially uploaded to temporary storage, enforcing a modest maximum size (because profile photos wouldn't be terribly large), and at least validating it as technically valid picture of one of the formats you want to support, such as JPEG and PNG, and perhaps enforcing a supported pixel size and possibly also running it through anti-malware scan. You might also want the uploads to be moderated by you or someone you trust before publishing them.

Technically, it'd be possible to generate a presigned URL for S3 or CloudFront to allow authenticated access to the underlying S3 bucket, but if you allowed them to upload files directly to where they would be accessible by other users, there'd be nothing to prevent them from maliciously uploading dangerous or even illegal content.

Since you're familiar with developing your application running on your EC2 instance, perhaps the easiest way would be for the files to be uploaded initially to the code you have running there, enforce the size limit (in bytes), and optionally put it through an anti-malware scan with ClamAV, for example. You could check the file for expected format and pixel size and perhaps also normalise its size with an image processing library, like ImageMagick, and perhaps also put it through a manual review process before allowing it to be shown to other users.

Once the picture would be validated, preprocessed, and optionally approved for publishing, you could upload it from your application on the EC2 instance to your S3 bucket, making it accessible via CloudFront.