I usually pray the Office of Readings together with Morning Prayer in the morning. I can't do this anymore because of time constraints, so I'm thinking about doing it at night - like right when I'm in bed, which is at around 10.

I know you can pray the Office of Readings anytime, and as a layperson I'm not really obligated to do anything, but what's the proper way to do this?

So say I decide to pray it at night, do I pray the Office for the next day (like is it considered a vigil - which I read somewhere hinted at), and so I begin with the invitatory psalm? Or can I pray it as is (starting with "God, come to my assistance etc" and the office for the day)?

You pray the OoR for the next day, and you do not use the invitatory. You do that before MP in the morning.

Ah, I've been doing it wrong then. For the past day or two I just assumed you prayed the day's office at night, until I read something that hinted that you pray it as a vigil, and even then I wasn't sure if you began with the invitatory. Good to know. Thank you!

Since OoR can be prayed at any point during the day, you can do it the way you have been, except that the invitatory always goes before the first office of the actual day.