Mid-infrared (mid-IR) observations are powerful in identifying heavily obscured active galactic nuclei (AGN) that have weak emission in other wavelengths. Data from the Mid-Infrared Instrument (MIRI) on board the James Webb Space Telescope provides an excellent opportunity to perform such studies. We take advantage of the MIRI imaging data from the Cosmic Evolution Early Release Science Survey to investigate the AGN population in the distant universe. We estimate the source properties of MIRI-selected objects by utilizing spectral energy distribution (SED) modeling, and classify them into star-forming galaxies (SFs), SF-AGN mixed objects, and AGN. The source numbers of these types are 433, 102, and 25, respectively, from four MIRI pointings covering ~9 arcmin2. The sample spans a redshift range of ≈0-5. We derive the median SEDs for all three source types, respectively, and publicly release them. The median MIRI SED of AGN is similar to the typical SEDs of hot dust-obscured galaxies and Seyfert 2s, for which the mid-IR SEDs are dominated by emission from AGN-heated hot dust. Based on our SED-fit results, we estimate the black hole accretion density (BHAD; i.e., total BH growth rate per comoving volume) as a function of redshift. At z < 3, the resulting BHAD agrees with the X-ray measurements in general. At z > 3, we identify a total of 27 AGN and SF-AGN mixed objects, leading to that our high-z BHAD is substantially higher than the X-ray results (~0.5 dex at z ≈ 3-5). This difference indicates MIRI can identify a large population of heavily obscured AGN missed by X-ray surveys at high redshifts.