What is the best way to open airway of an unresponsive victim with no suspect neck injury?

This question has many possible answers that depend on the skill level of the responder.

The average person on the street would be best to simply roll the patient into the recovery position, tilt the head back and clear any solid obstructions such as the tongue or vomit and angle the head to allow liquids to drain out of the mouth.

Artificial airways can be used if they are available and there are many to choose from, once again depending on responders skill level. simple oropharengeal or nasopharengeal airways are effective and are used in conjunction with intermittent positive pressure ventilation. laryngeal mask airways are harder to insert but achieve better delivery. the ultimate in airway management though, the gold standard, would have to be endotracheal intubation, where an ett tube is inserted through the vocal cords and delivers air directly to the left and right primary broncus.

Always bear in mind though, whilst suspecting a spinal injury is all well and good, maintaining a viable airway will always take priority over not wanting to move the head for fear of further damage. generally speaking, it's better to be alive and in a wheelchair then it is to be dead.