I only own Wings of Liberty. Is there a way for me to play the Whispers of Oblivion missions?
The 'Whispers of Oblivion' beta early-access missions are now accessible as the prologue to the main Legacy of the Void campaign. i.e. You will have to purchase Legacy of the Void if you wish to play them:
Upon clicking on the 'Campaign' button you will be taken to the screen to select your campaign:
When you go into the Protoss campaign, you will have the option of playing the Whispers of Oblivion Prologue missions: