Akram, who previously played for Essex and Northants, will not count as an overseas player after living in Britain for the past two years.
Wright joins from Leicestershire after making his Championship debut against Sussex last summer.
"I'm delighted with both signings," said Director of Cricket Peter Moores.
"Akram is coming into his prime as a fast bowler and he will give our attack a real spearhead. Luke still has a lot to work on in his game but he has genuine talent and the appetite to improve and work hard."
Akram played nine Tests and 23 One-day internationals for Pakistan, but has been out of favour since 2001.
He made four Championship appearances for Essex in 2003 after agreeing a short-term contract and picked up 20 wickets, including figures of 8-49 against Surrey.
"I could have stayed with Essex and Durham were also interested but once I spoke to Peter Moores I knew Sussex was the county for me," he said.
Sussex are still seeking a second overseas player for next summer, with South African batsman Gary Kirsten a possible candidate.
Pakistan leg-spinner Mushtaq Ahmed has agreed a two-year deal after taking 103 wickets last season.
Gavin Hamilton to play for Durham