What are the challenges for a submarine to move around in the deep sea?

"Deep Sea" for a submarine generally refers to areas of the ocean in which the depth is well below any operational limits, e.g, the middle of the Atlantic.

The biggest challenges for submarine deep ocean operations are:

1. Environmental / Acoustic Properties - Modern submarines are heavily reliant on the acoustic environment of the ocean area they travel in, and constantly monitor the ocean environment for changes in properties that can affect sound travel though the water (e.g., temperature, depth, etc.). Knowing where thermal layers are gives a submarine a distinct tactical advantage over other submarines and surface warships. Without constant monitoring of the surrounding environment and knowing how it affects incoming signals to the different Sonar arrays, the boat is at a tactical disadvantage at best, blind at the very worst.

2. Readiness / Training - when you're out in the big pond, seconds are critical in any casualty aboard ship. Any fire, flooding, or major accident must be dealt with swiftly or the risk of slipping below crush depth is a real possibility. Trust me when I say that "Flooding in the Torpedo Room" at Test Depth is something that you never want to hear - ever.

Believe it or not, it's navigating in shallower water that presents a bigger challenge. In the big pond, unless you're actively tracking someone or the environment is such that it's acoustically blinding (e.g., thermal layers prevent contacts from being heard until you reach a shallower depth), there's not much of a worry for moving around, even at deep submergence depths. Shallower water is a bigger problem because of underwater obstacles (both charted and uncharted). It has not been uncommon for modern submarines to collide with underwater mountains.