Karanaka was appointed at the City Ground on Monday.
He guided his former club Middlesborough to the top flight in the 2015-16 season but was sacked in March 2017.
The 44 year-old said: "After 10 months to come back is nice, and especially when you come back to a big club like Forest.
"I had a few [offers] in England and abroad but, if I had one thing clear when I left Middlesbrough, it is that I had to be really sure in the following step.
"Everybody wants to be in the Premier League. But I don't have any doubt that, working in the way we are going to work, we will be successful.
"When you have an ambitious ownership and you have an ambitious coach, you want to be successful. Now we have to transmit that ambition to the players."
Forest are currently 14th in the Championship table, and under caretaker Gary Brazil, knocked out holders of the FA Cup Arsenal this past weekend.