Well, in the old days, the clocks indeed had Chinese characters on them. Back then, they were called sundial instruments, or 日晷 in Chinese. Here is what this kind of clock looked like in the Forbidden City.
Perhaps with the passage of time, the characters on them have become blurred. Here is another sundial instrument, whose Chinese characters were much more clear.
Just like the modern clocks, there are 12 parts in the clock. These 12 Chinese characters were from the Twelve Earthly Branches, which are 子 (23:00–1:00), 丑 (1:00–3:00), 寅(3:00–5:00), 卯 (5:00–7:00), 辰 (7:00–9:00), 巳 (9:00–11:00), 午 (11:00–13:00), 未 (13:00–15:00), 申(15:00–17:00), 酉(17:00–19:00), 戌 (19:00–21:00) and 亥 (21:00–23:00).
As you noticed, in the Chinese version of counting time of the day, every period consists of two hours. Therefore, it is divided into two parts, which are 初 and 正. For example, if it is 子初, then it falls into the bracket of 23:00–0:00. If it is 子正, then it falls into the bracket of 0:00–1:00.