Alexander was battling second-year player Kony Ealy for the every-down end spot opposite Charles Johnson after being suspended for 14 games last season for violating the league's substance-abuse policy.
Alexander said he will undergo surgery Tuesday.
"I was stepping back, and it felt like somebody just stepped on the back of my leg," Alexander said as he stood in the locker room on crutches and a walking boot on his right ankle. "I looked back, and there wasn't nobody back there."
Alexander, who has failed multiple drug tests, came into the season thinking this was his last chance to succeed in the NFL.
He called the Achilles injury "a little bump in the road."
"I know God still has a plan for me," Alexander said. "My faith is still high. I'm not turning my back on God or anything. I'm going to still stay positive and focused."
Rivera said all three played better against the Dolphins after struggling last week against the Buffalo Bills.
"Those three guys in particular were exciting and fun to watch," Rivera said. "Frank was having a good camp and was battling for position and did some good things. That's a tough one."