In order to gain access to the Boutique in Lumiose City, all you have to do is partake in various activities inside Lumiose City. This could vary from simple battles in the streets, to getting a new hairstyle, getting your Furfrou a new hairstyle, making a PR video, eating at restaurants, helping out in the Lumiose Hotel, even taking a cab occasionally raises your style.
Just keep doing any or all of these things to increase your style until they allow you inside.