Backgrounds have a few purposes in 5e. From a character development standpoint, they define who they were prior to becoming Joe the Wizard or Bob the Fighter. In this way, they help to flesh out characters more fully, by helping to give them a place in the world they live in.
From a game mechanics standpoint, backgrounds give a character proficiencies appropriate to the background as well as some thematic, loosely defined ability that let's them excel in some non-combat manner.
To directly answer your question about changing a level 5 soldier's background, the short answer is not without DM's permission and essentially DM's fiat. This makes sense from both a story perspective and game mechanics perspective. For 5 levels now, that character has been presenting himself as a soldier, so it'd be odd if they suddenly were not. Also, for 5 levels, they have been using the skill proficiencies granted by having the soldier background.
If the character's goal is to shift their ideals, flaws, etc., there is some basis for that in both the PHB and DMG. These are underlying traits of a character that provide guidance on how the character interacts with the world, but they're not mandatory rules. On page 125 of the PHB, there is information to personalize a background if nothing is a perfect fit. It may be prudent to utilize this and consult with your DM if you want to change some of these aspects of your character. Also, there are periodically effects in the game which make your character gain a new flaw, but I suspect this may be beyond the scope of what you're asking.