How can I build a trellis?

I would build a wood planter box lined with plastic and attach a trellis to the back of it.

I would fasten a trellis or lattice to the uprights with something removable such as florists wire or nylon cord or rope.

I would buy a standard ready made piece of trellis either plastic or wood. It would look better if it had a frame, which would be easier to make out of wood. Unfortunately your picture does not show the base nor the top of the area, so it is hard to be more specific, It is a good idea to do this for beauty and softening.

You could use planter boxes with wires wound up the posts for the plant to climb. Another option is to speak to the landlord. I've found every landlord I've had has been keen to use me to improve their properties lol! It's possible she may like the concept enough to underwrite the costs with you donating the labor - that's what I've always proposed and it's worked!

these are more pics that might help show what im talking about

Make or buy a trellis, set it on a planter box.

https://www.hometalk.com/diy/grow/cedar-planters-with-trellises-3278127

https://www.hometalk.com/diy/outdoor/garden/diy-trellis-planter-8891983

https://www.hometalk.com/diy/build/planters/diy-raised-planter-with-trellis-9557565