A custom rifle has a ton of pro's. You get to pick the action, the stock, and the barrel. You can pick your own components such as the trigger and in some instances the firing pin assembly. You can build with a muzzle break or without one depending on the caliber. The sky is the limit with the custom rifle. You start from scratch and build something that most other hunters will never see in their gun cabinets.
The cons are wait time for the build (depending on builder and components availability), cost, and you will never get near what you put into it when and if you try to sell it. Cost can be as little as $1500 if you use a tuned factory action with aftermarket parts (trigger, action bedded and low cost stock, no optics), or as much and not stopping at $10,000. The average is probably in the neighborhood of $3500 with and without optics. With really good optics you are in the $5000 range and up.
To start you have to figure out how long you want to wait to save and what components you want exactly. You need to decide what caliber and what purpose you are exactly trying to achieve. Is it a hunter or a target? You state you want a hunter style stock, so in my opinion a Manners UltralightHunter
would fit the bill. You can choose as to how much or how little carbon fiber is built into the stock.
Then you have to decide what action. This should actually be the first step, but this also includes caliber. A medium bore caliber (.308, 243, 6.5x47, 6.5x284, 284 etc.) determines the size and length. If you want an overbore (7mmSTW, 300RUM, 338EDGE, 338 Lapua) then a larger generally more expensive reinforced action is required. This adds weight, but should you have a light stock then it is a mute point.
Then you choose barrel manufacturer. Lilja, Shilen, Douglas, Lothar Walther, Hart, just to name a few. These companies offer different variations in the same design of rifling, cost, groove count, and material make up. All are very good barrels and some are more expensive then others, but none really give up anything to the other. Application, application, application! Can't emphasize that concept enough deciding on the build options.
Trigger group is necessary as to what weight you want to pull each shot. Do you want an ultra sensitive trigger like a target 6-8oz or heavier (preferred) 1.5-3lbs. for hunting. The latter would be the proper application for a walking hunter. Then we move into bases for your optics. This is important for the action party because some actions of integral bases built into the action. So when you figure all of this out, then the scope comes into play, but I'll leave you to wrap your head around this info first.
P.S. $3500 should be the goal for rifle alone to save for!