You can put items into the cargohold from your inventory by pressing F5 to open the Hold window. You can then move items to and from your inventory, providing you are standing on a cargohold square of the ship when you do this.
Alternatively you can open a comm tab with your city and buy items which are then put directly in the hold, to do this you must be standing in the hold of a ship which is currently in the city you are trying to buy from. You first have to transfer funds to your ship using the F12 Governance window, On the spacecraft tab there is a section at the bottom to transfer funds to and from your ship. When purchasing from the city, your ship uses it's funds rather than yours. The 'ship funds' count does not seem to update properly, so don't press the transfer funds button more than you have to, because the funds are actually being transferred to the ship.
A current bug causes the cargohold to empty crates that are put in it, so try to avoid placing non-empty crates in the cargohold. Non-empty crates are not supposed to be able to be placed in cargo holds at all.