Call of Duty: Black Ops (Xbox 360) - No Split-Screen Support
Call of Duty: Black Ops (2010) does not support split-screen multiplayer on Xbox 360. You cannot play local co-op or versus with two players on one console.
Your Options
Online Multiplayer: Requires two consoles, two copies, two Xbox Live Gold accounts.
System Link (LAN): Connect two Xbox 360s via Ethernet for local network play (still needs two consoles/copies).
Later Black Ops Games Do Support Split-Screen
Black Ops II (2012)
Black Ops III (2015)
Black Ops 4 (2018) – limited modes
If local split-screen is a must, skip the first Black Ops and jump to a sequel.