Unless you are willing to do this correctly... I would not recommend using this water for drinking or "potable" use. How much rain do you actually get and is it really worth the cost? Do you do enough irrigation to use this water exclusively for that?
In general, a surface supply should have a redundant sanitization method. There are a multitude of ways to do this. Chlorine is the obvious choice. You should follow that with UV or you could also consider some form of properly rated filtration method.
Rain water tends to be very low tds/pH so correction can also be needed to protect your plumbing. A simple polyphosphate injection, soda ash, calcite or other form may be needed.
If you use if for irrigation... a simple filter will suffice so long as you don't drink from the sprinklers.
In Tucson, Arizona I used reclaimed 'grey' water from the city. For a residential use like mine at the time, the amount of paperwork and bureaucracy involved was enormous. I had to protect my Xeriscape drip irrigation from people's pets and wild animals too. The annual certification required is not something I would do today.
Where I live now, I have 40'x60' equipment shed I plan on using. After looking up how many gallons I get from 1000ft*ft of 'catchment', a 5,000-gallon tank may be a bit undersized.
I have two other areas a bit smaller and one a lot larger for possible 'catchment' too. I'm on 40 acres and not in a backyard so, during a typical 'dry' Summer I'm pretty sure I would drain it dry. If nothing else, I would use it in my front pond instead of well water.
After more thought about sanitization, I'm leaning real hard towards Chlorine. For Ph, I have very little concern adding stuff like I do now to fish ponds and tanks (>750-gallon industrial setups on solar power). In retrospect, for household water use, branching off my well to drive water with a pump off 330-gallon IBC tanks from my rainwater tank would suffice nicely for low well outputs if that becomes an issue during a dry spell.
A bit of chlorine and some other chemistry for Ph in an IBC plus what the well house currently does should protect both my health and my plumbing. I'm still kicking around my options though so, other thoughts and recommendations are appreciated and welcome!