It is unclear how long Lex had been keeping a journal, but it seems likely that this is something that he did later in his life, as he said early parts of his life before the first meteor shower were blocked from his memory and he only remembered these events after he was shot in Detroit.
Clark learned about Lex's journal during a flight with Tess on the way to a press conference. Clark reacted nervously to this news and attempted to get Tess drunk so she would reveal what the journal said. Clark asked her but Tess would not say. The contents of the journal are still unknown, however, Tess was able to discover all the information she needed from Lionel's journal.
Lionel had been part of a secret society called Veritas that was waiting for the traveler. He recorded this in a journal with writing and sketches. Tess found it after Lex's estate and business were handed over to her.
After reading the journal Tess quickly acts on the information and reveals the contents to Davis Bloome after she tried to kill him in an explosion. The journal explains that Davis was found by a couple of soldiers in a field after the meteor shower. Davis was then taken to the Luthor Mansion, where he had needles put in him to test if he was the traveler. Tess surmises that Lionel sent a search party for the Traveler and he didn't realize that there were two boys that fell to earth that day, the Traveler and Davis.
Despite not explicitly revealing that Clark was the traveler in the journal, Tess was able to work it out, due to the links with the Kents. The journal revealed that Davis' tests didn't match that of the Traveler's and after getting a phone call from Martha Kent asking for help with an adoption, Lionel threw Davis onto the street. This proves that Lionel had at least suspected that Clark was the traveler all along. Tess was only able to discover his secret through the information in the journal.