Hi and welcome!
(You’re in a similar boat to me back when I started weight training; I feel I have a lot to offer regarding this particular question, so although I’ll try and be as succinct as possible, feel free to ask anything else or for further explanation).
First off, I agree with 2 of your statements;
(1) you’re underweight (or at least, nowhere near your ‘natural muscle’ potential), and (2) weight-training, or weight-training style exercises are the best way to go about addressing (1) – at least to begin with (further detail below).
To put your ‘stats’ in context, I am 5’7” (168cm), 170lbs (75kg), and around 10-11% BF, with a BMI of around 28 (just to highlight the uselessness of BMI!). So were pretty much the same except I’m 50lbs heavier. It only took me around 9-12months in a gym to go from 60kg->75kg in the same lean state.
Rather than reeling off a list of vanilla (but largely true) statements like ‘do compound moves’, ‘keep intensity high’, ‘eat more calories’, I’ll point you towards some useful online/ paper resources, and invite you to do some reading:
If you take the advice of (what I assume) 99.9% of people will give you (i.e. hit the gym). I recommend you check out all the following resources even if you have no intention of joining a gym; the information around how/why/when to train, and how to best go about it is invaluable:
Specific training methods:
stronglifts.com (beginners training, recommended);
elitefts.com (& 531 ebook download $19.99);
book: SuperSquats by Randall Strossen (brutal, not recommend for at least 6 months, but by far-and-away the best training method for getting big and strong fast)
general resources (my list after years of sifting through rubbish):
book: Power to the People! By Pavel Tsatsouline
google search: ‘Greasing the groove Pavel Tsatsouline’ READ THIS!
while stronglifts 5x5 is a bit of a bland method of training, this is great for ‘gym newbies’ as it keeps the volume relatively high, enabling you to focus on muscle size, endurance, correct form, and keeps the strain off your joints and ligaments.
Jim Wendler’s 5-3-1 e-book is fantastic and my preferred method of training. Whether you use the system or not, the reading around assistance exercises, their purpose, and how many you need and when, is again invaluable.
If this is the route you decide to go, I also recommend asking this sort of question on a forum like sugdenbarbell, uk-muscle, or even myprotein (whilst studiously avoiding anything body-building.com related).
If you stick to ‘other activities’ and some home-exercise, good on you (probably healthier in the long run!), but progress will be considerably slower. If possible, I recommend you consider gymnastics.
The issue with gymnastics is it also requires a base-level of conditioning; but once you get there you'll get hooked. And things like planches, tucks, leg raises, human flags, hand-stand pushups et seq are, in my opinion, the ideal exercises for paleo-enthusiasts and general health.
All of the above is linked to STIMULATING muscle growth. As mentioned by plenty of others; to actually ACHEIVE muscle growth, simply stuff your face with more food. The more, the better, simple as. I'll leave the question of diet to the resident experts; since my answer would be to start drinking lots and lots of milk (paleo or not :P)
As a final thought: and yes I did promise I wouldn’t give generic recommendations, but I can’t stress enough the benefit of one particular exercise: deadlifting. Even at home, all it takes is a concrete floor, a few hundred kilos of Olympic plates, and a bar. Including deadlifting into whatever you choose, will increase you progress exponentially.
Hope some info linked here you find useful :)