According to such sources as Mark Eccles' exhaustive account, Shakespeare in Warwickshire (1961), there is no evidence other than some late and probably unreliable stories that Shakespeare ran away from home.
A story attributed to a William Castle, parish clerk at Stratford, asserts that Shakespeare had been bound apprentice to a butcher but had escaped from this by running away to London to become a player (story dated to 1693). However, this seems to be merely an expansion of an earlier account by John Aubrey that Shakespeare's father was a butcher, which is clearly incorrect.
Between 1688 and 1708, a tale became current that Shakespeare had had to flee to London because he had been caught stealing deer and rabbits from the park of Sir Thomas Lucy. This account was improved around 1703 by an alleged "bitter ballad" that Shakespeare was supposed to have written and posted on the gate of Lucy's park. The problem with these stories is that Sir Thomas Lucy didn't have a deer park in Stratford, and so they must all be late inventions.
Instead of running away from home, Shakespeare may have joined a company of traveling players that took him to London. Any concrete evidence of this is long lost, though.
Another possible source of conflict is Shakespeare's marriage in 1582. However, the details of this are often misinterpreted. His marriage was carried out in some haste and his wife was older than he was, but neither of these circumstances were particularly unusual. They may, for instance, have contracted a so-called "handfast" marriage that would have rendered any children born to them legitimate even without a further ceremony. Lingering hostility is alleged because of his later bequest of the "second-best" bed to his wife, but in fact that would have been the bed the couple had slept in, the best bed being reserved for guests.
In short, we cannot identify with certainty any conflicts in Shakespeare's personal or family life that might have affected his career or work, though as always, "absence of evidence is not evidence of absence."