Voldemort being an orphan stayed in an Muggle orphanage during the holidays. He disliked the orphanage and came to look upon Hogwarts as a home, just like Harry.
“My dear boy,” said Dipper kindly, “I cannot possibly let you stay at school over the summer. Surely you want to go home for the holidays?”
“No,” said Riddle at once. “I’d much rather stay at Hogwarts than go back to that — to that —”
Harry Potter and the Chamber of Secrets, Chapter 13, The Very Secret Diary
Sometime during his fifth year in Hogwarts, Voldemort learned of his Slytherin ancestry, who was one of the founders of Hogwarts, his desire to associate with Hogwarts might have increased.
When he was sixteen years, he left the orphanage to which he returned annually and set off to find his Gaunt relatives. He met his uncle Morfin Gaunt and came to know about Slytherin’s locket.
“Ar, he left her, and serve her right, marrying filth!” said Morfin, spitting on the floor again. “Robbed us, mind, before she ran off., where’s the locket, eh, where’s Slytherin’s locket?”
Harry Potter and the Half-Blood Prince, Chapter 17, A Sluggish Memory
This might have been the point at which Voldemort would have contemplate searching for all magical items associated with founders of Hogwarts. But I believe the final decision was taken when he was seventeen years old.
“He reached the seventh year of his schooling with, as you might have expected, top grades in every examination he had taken. All around him, his classmates were deciding which jobs they were to pursue once they had left Hogwarts. Nearly everybody expected spectacular things from Tom Riddle, prefect, Head Boy, winner of the Award for Special Services to the School. I know that several teachers, Professor Slughorn amongst them, suggested that he join the Ministry of Magic, offered to set up appointments, put him in touch with useful contacts. He refused all offers. The next thing the staff knew, Voldemort was working at Borgin and Burkes.”
“At Borgin and Burkes?” Harry repeated, stunned.
“At Borgin and Burkes,” repeated Dumbledore calmly. “I think you will see what attractions the place held for him when we have entered Hokey’s memory. But this was not Voldemort’s first choice of job. Hardly anyone knew of it at the time — I was one of the few in whom the then headmaster confided — but Voldemort first approached Professor Dippet and asked whether he could remain at Hogwarts as a teacher.”
“But he didn’t get the job, sir?”
“No, he did not. Professor Dippet told him that he was too young at eighteen, but invited him to reapply in a few years, if he still wished to teach.”
“How did you feel about that, sir?” asked Harry hesitantly.
“Deeply uneasy,” said Dumbledore. “I had advised Armando against the appointment — I did not give the reasons I have given you, for Professor Dippet was very fond of Voldemort and convinced of his honesty. But I did not want Lord Voldemort back at this school, and especially not in a position of power.”
Harry Potter and the Half-Blood Prince, Chapter 23, Horcruxes
So after completing his education at Hogwarts, Voldemort had two plans:
- Start teaching at Hogwarts
- Search for magical items associated with founders of Hogwarts
His plan number 1 failed, he decided to work on plan number 2. It was during his work at Borgin and Burkes he came across Hufflepuff's Cup.
So by the time Voldemort was 15-16 years old, he had planned to use magical items associated with founder of Hogwarts. He decided to go through with his plans when he was 17 years old.