"Dolgozik" is one of the -ik verbs, so in first, second and third person singular it gets the suffixes -m, -l, -ik, respectively: én dolgozom, te dolgozol, ő dolgozik.
Also, the stem is "dolgoz-", so the correct form in second person plural is "dolgoztok".
Edit: Apparently no, the -l suffix is not because of the -ik conjugation, but purely because the stem ends in -z.
I believe the -l ending has nothing to do with -ik and is related only to the -z ending.
Most -ik verbs end in one of the triggers for -l (z dz s sz), but e.g. törik is -ik but not z dz s sz and so has the regular törsz rather than töröl. And conversely, e.g. mos is not but -ik but is z dz s sz and so it has mosol, even though it has mosok mos and not mosom mosik.
You are right that if "dolgozik" were not an -ik verb, it would still get the -l suffix in second person singular because the stem ends in -z, but it wouldn't explain the -m in first person.
Also, "törik" is a so-called "pseudo -ik verb", meaning that it only gets the -ik ending in third person. Similar verbs are for example "folyik" (to flow), "(el)bújik" (to hide), "megjelenik" (to appear), "hazudik" (to tell a lie), "születik" (to be born), "válik" (to become or to divorce).
Finally I found evidence for what you're saying. Most resources claim that pure -ik verbs get -m, -l, -ik and that's all, and as a native I don't think twice before using these verbs, so I didn't notice the inconsistency. But after searching around a bit I found some sites that explain it fully and clearly.
It seems that originally it was indeed a characteristic of -ik verbs to use -l in second person singular, and all other verbs got -sz (for example, originally "olvasosz" was the correct form instead of "olvasol"). Later the conjugation merged in second person: non -ik verbs with stems ending in -dz, -s, -sz, -z started taking the -ik verbs' -l suffix, while -ik verbs with a different stem started taking the non -ik verbs' -sz suffix, leading to the rule we have today.
There can be differences in dialects (I often hear "te lakol" in the countryside), but basically yes, second person conjugation has nothing to do with -ik. I edited my comment above.