I hate to say it, but I think the answer is "with difficulty".
From my own experience, it's really hard to know what's bad, what's good and what's an acceptable workaround if you've never seen anything different. Myself, I got lucky and ended up working on a project after the start of my career with someone who could explain the whats and (more importantly) the whys of bad/good/ugly code bases.
Generally, try and get some skill in being able to view a codebase from a high level. Draw it out on a whiteboard in boxes. Perhaps do this on other, pet projects first as it's nearly impossible to do this with a spaghetti-code project. If you can't pick out modular parts, then you have a big ball of mud. If you can, try and work on making and keeping them uncoupled. If you can, try and work on finding the natural boundaries of the other code you couldn't break up, and make those less coupled (you don't need to solve the coupling problems all at once!).
Are there a mix of architectural patterns in the code? This is pretty common when you're working on a legacy project. It's what happens when you get someone who doesn't really know how to architect, or there were a bunch of folks throughout the history of the project who (probably) had the right intentions, but didn't get it finished. Or, and this is the worst, you had two or more team members trying to bend the project to their own preferences without communicating with each other. If this is the case, talk to your team, agree on one and then you can work towards getting the style consistant. You don't even need to pick the best one. Getting a project into a consistant state is better than having an ugly mix and match.
Are there a bunch of mixed up design patterns floating around? Try and refactor those out as much as possible. Design patterns are great, and you should use them where appropriate. But if you find a lot of them nested within each other, it's not a good sign and probably indicates someone at some point swallowed a design pattern book and thought it would be a good idea to implement them. All of them. Nested patterns can more the likely be refactored out to simplify the code. Though again, make sure you understand what they are there for first. Otherwise you may be unpicking something intentionally complex that needs to exist to remove complexity elsewhere.
What does the DB look like? Is it designed around the projects business logic? Is this sensible for your project? Personally, I dislike putting any business logic into the data storage layer but it might be sensible for your particular project, so YMMV. If business logic in the DB is causing nasty workarounds, then you may have something else to refactor there, though this may not be possible.
Never refactor just for the sake of it! If you don't have buy-in for your ideas on how to improve a code-base from the rest of your team, you're going to be creating problems. You may also be missing critical information that your tech-lead knows about and made design decisions based on it. There have been several times I've tried to make things better as a Junior dev, only to find out I'd made some bad assumptions and created a mess.
Don't refactor without tests either. The system may be reliant on strange code, so make passing tests before changing things. That way you at least know the behaviour hasn't changed.