死死的 is an adverb formed by reduplicating 死 (sǐ) and adding 的; it modifies verbs to mean doing something with extreme force, fixation, or determination — e.g., holding tightly, staring fixedly, or remembering firmly. It always appears before the verb in Chinese, and the 的 can be interchanged with 地 in formal writing (though 的 is more common in colloquial speech).