The 30-year-old sustained the injury in the second quarter of his side's 37-34 loss at Philadelphia Eagles and Hall is already thinking about the road to recovery.
"I'm not a spring chicken, so I got to make sure I get it done the right way," he said.
"I've got to be as patient with it as possible, attack this rehab and get back out there on the field.
"I just thought I got stepped on. I didn't know instantly. I didn't think much of it. I thought I had a calf strain or something."
Hall revealed he has spoken to Los Angeles Lakers star Kobe Bryant, who suffered a similar injury in April last year.
"He was probably the most notable guy that I remember having an Achilles injury and coming back," he said.
"I talked to him for a long time yesterday about it and he said the same thing our trainer said which is good. I'm getting the same information, just to be patient with it.
"It's going to feel like you can push, it's going to feel like you can do some things, but the best thing is just to not do anything.
"This being injured thing is kind of new for me. I'm going to heed the advice of all these people, and I'm going to listen to them and I'm going to come back better."
Redskins head coach Jay Gruden said that rookie Bashaud Breeland would be the likely candidate to replace Hall, whom he said would be missed both on and off the field.
"He was voted captain for a reason," he said. "Guys look up to him. He's a veteran guy and has that experience that you can't coach.
"He communicates well with the young guys and is a great leader by example for the young guys like Breeland and [David] Amerson, and unfortunately, he's going to be gone."