I'd say build one unless budget is the main concern. You can get a Bergara, RPR, or Savage for around $1000 for the whole gun. Building one will start around $2000 and go up depending on what parts you pick. But, even if you keep it down around $2k, you will have a very nice rifle that will handle and shoot very well.
There are several great actions available now for around $800 to $1000. This wasn't the case a few years ago... used to have to spend a lot more for a custom action.
After researching them all out, I ended up going with a Nucleus action from American Rifle Company (ARC). It was just released a year ago and is a very nice action for the price. PM me if you want more details.
Start your custom rifle build by picking the action. Highly recommend a Remington 700 clone (ie: Nucleus, etc) in a short action. This will give you the most options for your other parts (stock/chassis, trigger, etc).
Also recommend the action you pick is threaded for Savage small shank threads. (Nucleus, Origin, and others are) This will allow you to buy readily available Savage style pre-fit barrels. (Search for internet videos on headspacing a Savage style barrel with barrel nut).
If you have put together an AR, then you will have no problem putting together a bolt action custom.
Lots of help here if you decide to go the custom route!