The 8-Bit Guy on Youtube made a great video series on character LCD screens and how to run them. In the first video he shows how to drive one manually using just switches: [this video] In parts 2 and 3 he shows two ways how to drive them with a C64: [link] [link]
If you don't need actual HDMI and a regular monitor then there are other solutions which are more like what you describe: Firstly there are simple character LCDs which all have a standardized and simple electrical interface[1] for sending ASCII characters and control codes. Then there are bit-addressable colour displays that let you send drawing commands, eg: [2] [3] [4] [1] [this video] [2] [link] [3] [link] [4] [link]
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