Aight, did a bit of Googling and searching here, the only question I found related was this, although the only answer it had wasn't marked as accepted, is old and is confusing.

My problem is basically what I've said in the title. What happens is that the GUI freezes while the upload is in progress. My code:

// stuff above snipped

public partial class Form1 : Form
{
    WebClient wcUploader = new WebClient();

    public Form1()
    {
        InitializeComponent();

        wcUploader.UploadFileCompleted += new UploadFileCompletedEventHandler(UploadFileCompletedCallback);
        wcUploader.UploadProgressChanged += new UploadProgressChangedEventHandler(UploadProgressCallback);
    }

    private void button1_Click(object sender, EventArgs e)
    {
        if (openFileDialog1.ShowDialog() == DialogResult.OK)
        {
            string toUpload = openFileDialog1.FileName;
            wcUploader.UploadFileAsync(new Uri("http://anyhub.net/api/upload"), "POST", toUpload);
        }
    }

    void UploadFileCompletedCallback(object sender, UploadFileCompletedEventArgs e)
    {
        textBox1.Text = System.Text.Encoding.UTF8.GetString(e.Result);
    }

    void UploadProgressCallback(object sender, UploadProgressChangedEventArgs e)
    {
        textBox1.Text = (string)e.UserState + "\n\n"
            + "Uploaded " + e.BytesSent + "/" + e.TotalBytesToSend + "b (" + e.ProgressPercentage + "%)";
    }
}