Here's the working code for doing the multipart upload.
public UploadPartResponse UploadChunk(Stream stream, string fileName, string uploadId, List<PartETag> eTags, int partNumber, bool lastPart)
{
    stream.Position = 0;
    //Step 1: build and send a multi upload request
    if (partNumber == 1)
    {
        var initiateRequest = new InitiateMultipartUploadRequest
        {
            BucketName = _settings.Bucket,
            Key = fileName
        };
        var initResponse = _s3Client.InitiateMultipartUpload(initiateRequest);
        uploadId = initResponse.UploadId;
    }
    //Step 2: upload each chunk (this is run for every chunk unlike the other steps which are run once)
    var uploadRequest = new UploadPartRequest
                        {
                            BucketName = _settings.Bucket,
                            Key = fileName,
                            UploadId = uploadId,
                            PartNumber = partNumber,
                            InputStream = stream,
                            IsLastPart = lastPart,
                            PartSize = stream.Length
                        };
    var response = _s3Client.UploadPart(uploadRequest);
    //Step 3: build and send the multipart complete request
    if (lastPart)
    {
        eTags.Add(new PartETag
        {
            PartNumber = partNumber,
            ETag = response.ETag
        });
        var completeRequest = new CompleteMultipartUploadRequest
        {
            BucketName = _settings.Bucket,
            Key = fileName,
            UploadId = uploadId,
            PartETags = eTags
        };
        try
        {
            _s3Client.CompleteMultipartUpload(completeRequest);
        }
        catch
        {
            //do some logging and return null response
            return null;
        }
    }
    response.ResponseMetadata.Metadata["uploadid"] = uploadRequest.UploadId;
    return response;
}
This will help you implement a chunked file upload to S3.