I don't think anyone can give you a budget recommendation because what you'll need to invest depends on the niche you're in (competitive niches like finance, web hosting, etc. require more money than other niches), and the keywords you're targeting.
If your domain name is new you should start off slowly, with only several backlinks per month, all of them directed to your homepage. I would say go for 2-5 backlinks in the first month after the site goes live (and indexed in google, obviously) but it's equally important to continue posting content regularly during your first 6 months. A few articles per week should do the trick along with 2-5 backlinks per month, like I said, pointing to the homepage.
However - and this is very important - do NOT buy shitty backlinks for a new domain name. I have done the same with one of my domains and now I don't know how to improve its ranking without digging deep into the pocket. If your domain name is new it's very important that you only give it quality backlinks, in the form of guest posts on quality websites (sites with organic traffic and low OBLs), or in the form of PBN backlinks but from quality PBNs again (PBNs that are indexed in google, that have no penalties of any kind, and which also have a low number of OBLs).
Obviously that getting such backlinks will cost you more than your typical SEO package that costs $45 (just an example), and delivers "quality backlinks from high DA sites". You are probably looking at several 100s of USD for 5-6 such quality backlinks, but if you care about your site and you have the money go for it! It will be worth it!
Also, build 10-20 quality backlinks to the homepage and keep an eye on your site's articles (pages / posts) to see which one gets picked up by google. Many times, you will see pages pop up in top 100 rankings out of the blue, pages that you would have never thought about ranking anywhere because competition seemed fierce for that keyword (it's currently happening to one of my websites, where the only keyword ranking in top 100 (on position 41 more specifically) is one of the 3 keywords that my keyword research tool says it has "fierce competition", while all others are easy / very easy to rank for, but they're nowhere in google LOL).
So anyway, long story short, make sure you only get backlinks to your homepage until one of the inner pages starts showing up in the rankings, and when it does start (slowly again, this is very important) building backlinks to that particular page to boost its ranking. One, or two backlinks per month should do the trick for that particular inner page, maybe some social signals, too, but make sure they're all quality backlinks and social shares from aged accounts with active followership.
If your domain is aged (more than 1 year old) you should be safe with building more than a few backlinks per month, depending on its existing backlinking profile, but this will obviously cost you more (probably in the 1000s if you can't find a good deal here on BHW)
