The first step, as people have elaborated below, is combust the compound and measure the weights of various oxides, which (after the atomic masses of the relevant elements were settled around the 1820s) lets you work out the empirical formula of an unknown molecule. For benzene, this would tell you that there is 1 C : 1 H, but this doesn't tell you if it's C₄H₄ or C₆H₆ or C₁₁₁H₁₁₁.
The second step is to determine the molar mass of your compound, which requires finding something that depends on the amount of substance but not the mass directly. (In modern times, this is primarily mass spec). Back in the 19th century, this is probably abusing the ideal gas law, which lets you compute the number of moles in a gas given the pressure, temperature, and volume of a vessel. Combine this with the mass of that container, and you know how much a mole weighs. If you get out, say, 77g/mol, and you know that the ratio is 1 C : 1 H, well, the only formula that makes sense is C₆H₆ (which should ideally have 78g/mol, but you might not get the right answer for various experimental reasons).
https://en.wikipedia.org/wiki/Argon#History
https://www.youtube.com/watch?v=loqudG71uBM (Chemistorian, noble gases)
I believe it's kali from German Kaliumhydroxid[0] (KOH, what it uses to dissolve CO2), from the same "potassium" root as al-kali in English, from medieval Arabic[1]. (And also metonymically a name for the coastal salt-marsh plant[2] from which medieval workers sourced potash/potassium[3]. I actually submitted that plant to HN [4] a few days ago, but no one was excited about it. They were once an essential ingredient in glassmaking, hence their other name, "glasswort").
[0] https://de.wikipedia.org/wiki/Kaliumhydroxid
[1] https://en.wiktionary.org/wiki/kali#English
[2] https://en.wikipedia.org/wiki/Salsola_kali
Strangely enough, the modern German name is Fünf-Kugel-Apparat, "five balls apparatus". I found that one simply by going through "Other languages" on Wikipedia.
And benzene is called Benzol in German. And gasoline is called Benzin - that word has false friend potential because it seems more similar to benzene. It is also not derived from the name of Carl Benz of Mercedes-Benz fame who used it in the first practical automobile that he invented.
How were chemists in the early 19th century able to determine benzene must be highly unsaturated without knowing its structure? Did they simply combust it and measure the amount of water vapor and carbon dioxide produced?