Prepare the ice cream base according to your recipe's instructions. You'll want to follow the recipe until the churning step, then continue with the instructions below:
Making Ice Cream Without an Ice Cream Machine
Chill ice cream base in the fridge until cold.
Transfer ice cream base to a freezer-safe container and freeze overnight, or until solid (see Note 1, below).
Once fully frozen, remove the ice cream base from the container and place into a high-powered food processor or blender (see Note 2, below). You may need to chop the ice cream into smaller pieces to fit into your appliance. Blend until smooth (see Note 3, below).
If there are any mix-ins (cookie pieces, nuts, etc.) in your chosen recipe, mix them in now.
Enjoy at this step for a softer ice cream or transfer to an air-tight, freezer-safe container to be scooped and enjoyed at a later time.
Notes
I prefer to use a gallon-sized Ziploc plastic bag at this step. It allows you to freeze the ice cream base with minimal air exposure and allows for easy removal. You can also choose to use an ice cube tray, but you'll want to cover it tightly to minimize air exposure - air exposure will form excess ice crystals.
If using a blender to blend the ice cream, I would recommend a blender with a tamper. That will allow you to push the ice cream blocks towards the blade as it spins instead of stopping and starting the blender several times to push the mixture down.
When it comes to blending the frozen ice cream, you want to work quickly. You want to blend the ice cream without allowing it too much time to melt. You may have to periodically stop blending to push the frozen chunks down towards the blades before resuming.
While this method does work well, please note that the resulting ice cream may be a bit more icy than if you were to use an ice cream machine.