Shevelen8 is correct about 'that', it is not needed in speech or in writing, but you can use it if you like.
Mohamed is right that your word order is unusual - it is correct but not the way most native speakers would say/write it. His alternative is good, but you can make a smaller/simpler change to your original text: "I didn't expect that they were both going to die".
Finally, your sentence uses a longer construction than necessary: English speakers are lazy and usually say things in fewer words:
I didn't expect that they would both die.
I didn't expect both of them to die.
I didn't expect them both to die.