How do you analyse and evaluate a primary source?

Analysing a primary source is, essentially, the same for any period of history. When you analyse and evaluate a source, you're really asking how useful it is. Therefore, we need to first define what "useful" means. A useful source is one that contains relevant information for a historical enquiry. The facts shown in the source will be accurate, and the author of the source will be reliable.
So how do we work out if a source is accurate and reliable?
You need to use your contextual knowledge of the period: does the source agree with the historical facts that you know are true, or does it say something different? If you can show that a source is factually correct, then you're able to analyse it and demonstrate its usefulness.
However, to fully evaluate a source, we need to think about how reliable it is. This is where you need to think about the provenance of the source - where a source has come from. Who is the author of the source, and what message are they trying to send? Were they present at the event they're portraying? (If they were, the source will be more reliable.) Do they have a hidden agenda - are they twisting the facts in order to persuade their audience? For example, a newspaper article might be very reliable: it was written at the time to present factual information, not opinions. A cartoon might not be reliable, however: although it might have been created at the time, it might have been created for humorous purposes, to belittle someone or something, etc. - it might not be showing the facts.
We don't always know the exact origins of a source, especially anonymous sources from the early modern period and before. This doesn't mean they're not useful: you need to fact-check them for accuracy. Just because a source is factually correct, it's not necessarily reliable, and vice versa. You need to look at accuracy and reliability together.