Uploading a file into Umbraco 8 has changed slightly, using the code below as found on the Umbraco Help Forums, worked for me:

Now you need the IContentTypeBaseServiceProvider and a Stream.

// Umbraco 8
using Umbraco.Core; <- Important..

// Needed Interfaces
IMediaService _mediaService; 
IContentTypeBaseServiceProvider _contentTypeBaseServiceProvider;

// The new way to do it.
IMedia media = _mediaService.CreateMediaWithIdentity("Filename", 1234, "File");
media.SetValue(_contentTypeBaseServiceProvider, "umbracoFile", "FilenameWithExtension", Stream);
_mediaService.Save(media);

Byte[]

byte[] byteArray;

using (Stream stream = new MemoryStream(byteArray))
{
media.SetValue(_contentTypeBaseServiceProvider, "umbracoFile", "FilenameWithExtension", stream);
}

HttpPostedFileBase

HttpPostedFileBase httpPostedFileBase;

media.SetValue(_contentTypeBaseServiceProvider, "umbracoFile", httpPostedFileBase.FileName, httpPostedFileBase.InputStream);

Thanks to Bo Jacobsen for sharing the solution!