While Diablo IV does have some set limitations on how many Healing Potions players can use, you will also be able to upgrade them through the course of the game. This can be done using crafting materials you find while exploring Sanctuary and by leveling up. Once you have access to Kyovashad and are level 10, you can immediately upgrade your potions so long as you have the right materials.
Thankfully, all you need are Biteberries, Gallowvine, and other herbs you can obtain through opening chests you are given by completing Side Quests. You will have more than enough materials come level 10 for that first upgrade, which will provide a decent boost to your overall healing. That said, you cannot upgrade your healing potions to max immediately. Instead, you will be gated by levels. So while you may have the right materials, until you reach the required level you cannot upgrade your Healing Potions in Diablo IV without that prerequisite being met.
You can see the level requirements to upgrade your Healing Potions below.