How can Daniel Hardman come back, against Pearson's wishes, 5 years after resigning?

The firm was still called "Pearson Hardman" in seasons 1 and 2, which implies that Hardman was still a partner in the firm, just a silent one (ie: his resignation was from active duty only.) Presumably, Jessica Pearson didn't have enough money to buy out Hardman's stake in the firm by herself, hence why they had to resort to blackmail in the first place.

When he came back in season 2, IIRC, Hardman said he had told his wife everything, so the blackmail threat was now void. He didn't need to be re-hired because he still owned half the firm and (presumably) had enough friends left on the board to overrule Pearson in the case of a vote on the matter. (I don't recall if that bit was addressed specifically in the show, but Hardman certainly had enough friends on the board to threaten Pearson's control in other issues that were raised later on.)

Source: Suits Wikia

Daniel Hardman, was the co-founder of Pearson-Hardman. He is an attorney and was the former managing partner before Jessica Pearson and Harvey Specter threatened to expose his affair to Alicia Hardman, his wife, who had cancer.

In the year 2007, Jessica had noticed someone had been stealing from the firm after looking at the accounting books. Hesitant to go to Daniel without any concrete proof, Jessica sends Harvey to investigate, promising him partnership if he succeeds. Harvey soon finds the culprit to be Louis, only to find out that the real culprit had made Louis out to be the fall guy. With Louis's help, Harvey proves that Daniel Hardman had been the one stealing from the firm. When confronted about this by Jessica, Daniel reveals that his wife has cancer and he has been stealing this money to help her. This was enough to satisfy Jessica but not Harvey who really wanted his partnership. He soon discovered that the money was not for Alicia but for Monica Eton, with whom Daniel was having an affair. Threatening to tell his wife, Harvey asks Daniel to sign a piece of paper saying Daniel is walking away from the firm. Daniel reluctantly agrees. And this was removing Hardman from actively participating firm's business and just being a guy whom name kept as a name partner, because Jessica couldn't came up with money to buy his shares.

First attempt:

Five years later, when Alicia loses the fight against her cancer, Hardman comes back and manages to take over the firm once again and takes advantage of a lawsuit filed against Harvey by Travis Tanner, saying Harvey had buried evidence in a case. Harvey, who had done no such thing, ordered Donna to find any such evidence in the file room. When Donna found the evidence stamped with her name on it, she destroyed it in an attempt to protect Harvey. Later, it was revealed that Hardman had planted the CM memo to make it look like Donna had buried it, and then had Travis Tanner attempt to sue the firm. Hardman was then dismissed from the firm for fraudulently suing the firm.

Second attempt :

More than two years later after his dismissal, Hardman once again returns to New York and is trying to take over the firm from Jessica and Harvey, by using senior partner Jack Soloff.