The D&D Adventurers League DM Pack has the documentation to answer what you're asking, and a whole lot more.
The biggest thing to be aware of is that it has rules that trump the guidance contained in older modules - there is a different table that is used now.
When you earn it, the entire lump (XP, GP, DT) is recorded as a single bundle. When you apply it, the whole bundle must be applied to a single character. Each award is applied individually, and only applies to one character.
As far as logging it goes, as long as it's clear, it's really up to you. Personally, I keep my DM Rewards in an Microsoft Excel workbook. It has columns for where the awards came from, what the amounts were, when I earned them, and most importantly which character I applied them to.
On the character itself, the same rule of thumb applies - as long as it's clear, it's all good. I log mine like an adventure, called "DM Rewards". I put the numbers where I normally do, and have a notes section where I include where the award came from. You can lump multiple rewards into one "DM Rewards" entry, as long as there is no actual play time before and after. Each entry should have a date, indicating when the award was applied, just like one indicating when the module was played.
The other thing you'll find in that package is the DM Quests. These are separate from the basic DM rewards just from running, giving you bonus rewards for things like running particular groups of content, running the same thing more than once, and so on. They modify and/or supplement the DM rewards.
These rewards change each season, so make sure you're always using the right season's version of the guidance on them. This is the biggest reason I keep track of the date I run things - it matters for which season they apply to.