Alcohol such as rubbing alcohol can kill bed bugs and the bed bug eggs.
To kill the bed bugs with rubbing alcohol just put some rubbing alcohol undiluted into a spray bottle and spray the bed bugs with the rubbing alcohol.
Spray any other areas with the rubbing alcohol too wherever you think there might be a bed bug infestation.
Diluting some tea tree oil into some water can also kill bed bugs and the bed bug eggs.
If you have a severe infestation of bed bugs though it's best to contact and hire an exterminator who is very good at getting rid of the complete bed bug infestation.
But doing those home remedies can help to kill some bed bugs and the bed bug eggs if it's not too bad of an infestation.