A Freesync compatible, IPS panel, 1080p monitor will set you back right around your budgets upper-end.
https://uk.pcpartpicker.com/product/rvjWGX/lg-23mp68vq-p-230-60hz-monitor-23mp68vq-p
144quid.
Display size, panel-type and response time would be the biggest factors IMO. Refresh rate, if you want to push some higher frames.
FreeSync is certainly an added bonus.
For example, a 4 or 5ms, IPS panel 60Hz isn't going to set you back much more than 100quid.
https://uk.pcpartpicker.com/product/dTvZxr/acer-r221q-215-1920x1080-60hz-monitor-umwr1ee001
Dropping to a TN panel can get you greater refresh rates.
For example, 1080p 144Hz for ~130quid
https://uk.pcpartpicker.com/product/ZBZ2FT/aoc-monitor-g2460pf
All depends on the titles you play (IMO). Are you looking for high refresh rates, or better picture quality?