The game server is not open to players yet.
Combat Achievements on ONYX are stock Old School tasks. The Combat Achievements entry in the character summary panel opens the interface, where you can browse tasks, bosses, and rewards.
The server records a task when the fight it belongs to can measure it. Kill-count tasks are recorded for most bosses the server already fights. Some other tasks are recorded when the server already tracks the extra condition in that boss fight, such as a kill time or a restriction. A task whose boss or activity is missing from the server is left unrecorded.
Each recorded task is worth the stock points for its tier: 1 for an easy task, 2 for medium, 3 for hard, 4 for elite, 5 for master, and 6 for grandmaster. You reach a tier when your total points are at least the combined points of every task the server currently records, from easy up through that tier. Points from any tier count. The official Old School list is longer, so the two point totals differ. Adding a recorded task later raises the ONYX total.
Ghommal, at the Warriors' Guild, explains the six tiers. Once a tier is reached, he can give you that tier's antique lamp and Ghommal's hilt. His dialogue also mentions other reward effects. Not all of them are confirmed to work on the server, so they aren't listed here.
The twelve fights with drop tables on this site are on the boss drops pages. A shorter account of what the server contains is on what's in the game.