The 26-year-old winger will be a Blues player next season after Dutch side Ajax agreed an offer of 40 million euros ($43.4 million) from the Premier League side on Thursday.
The Moroccan international lost his father aged 10, two of his older brothers served prison time for burglary and he had drugs-related problems with the police as a teenager.
Lampard, though, said Ziyech had left his past behind him in the performances he had seen including his outstanding display at Stamford Bridge this season in the 4-4 Champions League group stage clash.
"I had been watching him a lot," Lampard said at a press conference.
"I was impressed by him at Tottenham last season (Champions League semi-final). We want to bring in good people and he ticks a lot of boxes. I'm delighted.
"I have spoken to him. I know the story, his upbringing and home life. I would say from the outside when you see someone come through tough times, he has shown his quality on the pitch and attitude and that is good.
"I am excited to work with him."
Lampard's side face Manchester United on Monday in a pivotal match regarding the fourth and final Champions League spot with his outfit in fourth spot, six points clear of eighth-placed United.
The former Blues midfielder said he hoped Ziyech would provide a killer final ball which the team have lacked this term.
"The ability to make assists and score goals," he said.
"We have had a lot of the ball but not the final ball, hopefully he can bring that next season.
"He will have to fit in the way we want to play here, but the quality will hopefully be there to see."
Lampard refused to say if goalkeeper Kepa Arrizabalaga would regain his place for the United match.
Lampard had dropped the Spain international for their last Premier League game prior to the mini-break -- the 2-2 draw with Leicester City -- and brought in veteran Willy Caballero.
"The reaction (from Arrizabalaga) has been good, as I would expect," he said.
"I haven't had lots of conversations with him. It's a decision I will have to make. I don't take into account the fee (that Chelsea paid for Kepa). It's the training that counts.
"I know how it will feel for him. I had it in my career, it is part and parcel. I deal with the players as it feels right. I am open and straight with them."