Alchemical Weapons are a category of gear that is available to any character, not just Alchemists.
To clarify, the Alchemist class was introduced in the Advanced Player's Guide (APG), however prior to that alchemical items, such as Alchemist's Fire, existed in the Core Rulebook under the category of "Special Substances and Items" in the equipment section, which essentially consisted of items that Craft (Alchemy) could make.
As a preface to the section the items are contained in, it was written:
Beyond armor and weapons, a character can carry a whole variety of gear, from rations (to sustain him on long travels), to rope (which is useful in countless circumstances). Most of the common gear carried by adventurers is summarized on Table 6–9.
Given this preface, these items are definitely intended for the common adventurer to have and use, not just alchemists.
To explain how they became known as a Alchemical Weapons.
When Paizo released Ultimate Equipment, they republished a lot of the items from previous books including alchemical ones, in this book they decided to classify items crafted via Craft (Alchemy) for the intent of harming others as Alchemical Weapons. Thus, giving us the category of equipment we now know as Alchemical Weapons.