Incoming wall of text:
Remember that skills are capped by their related attributes, which are increased by gaining juice. So a brand new recruit with an intelligence of 3 might end up with an intelligence of 7 after doing enough illegal stuff, whereas you might be able to find someone with an intelligence of 9 to start with who ends up closer to 17 after building up juice. The first guy would have a max computers and law skill (as well as any other INT-based skills) of 9, while the second guy would have a cap of 17 in these same skills (which is so high that some skills won't get there without quite a bit of work, hacking only brings computer skill to 15 iirc).
As for how to play, that's up to you. The standard way to play is to grab a few people at the start and train them up with gun skills, while using hackers to get money to supply your efforts. You can also free some kids from the clothes factory/sweatshop in the industrial district, get their juice up a bit, and use them to make clothes, which you can sell for a profit. There's really a lot of choices as for how to get money. From there you can do what you want; you can try to have tons of death squads going out as much as possible, you can train a small ninja squad with huge disguise and stealth skill that can sneak its way past anything, or you might just try an all-legal playthrough and never lift a finger against anyone. For early playthroughs where you're still learning, the best advice is to never keep all of your important people in one place. Personally, I have a whole safehouse for just my founder, one for hackers, one for tailor/first aid/utility person, then use the rest of my safehouses for my murdering/fighting squads (they go between them so as to not have massive heat all the time and not get raided).
You can always shoot up the police station a bunch to get it closed down and prevent raids. Sleepers at the police station (acquired from recruiting, kidnapping, or seducing) can also warn you of incoming raids. Remember that going on murdering sprees will raise the heat of your safehouse, so change them often if you keep killing people. You can buy apartments in each of the different parts of the city, and they have varying costs. You can also upgrade your industrial district safehouses to have business fronts which will make your heat rise more slowly, as well as have defenses in the event of a siege.
If you want to play the legal system side of the game, try getting some sleeper judges and lawyers by recruiting, kidnapping, or seducing. Sleeper judges have various effects like making the jury more lopsided in your favor, and always granting leniency when found guilty. Sleeper lawyers just give you a free lawyer for when someone goes to court, with better stats than taking a state-appointed lawyer. If you have sleepers in the legal system and you're at risk of some important liberals dying, just have them surrender and take your chances in court. Better to give them a chance to live in court than have them fighting police officers while there's no police brutality laws.
The main goal is to sway the issues. There are elections every November 1st in even years only, so changing public opinion shortly before then can do wonders for the election results. Just don't wait too long, or you won't be able to get many issues swayed. The best way to change the public's stance is to raid related sites and just kill people. For example, if people don't want liberal judges, go raid the courthouses to raise awareness. If people don't care about sweatshops, raid the businesses in the industrial district. Just keep in mind that harming innocents during raids (by accident) can have the reverse effect, you alienate the masses and they won't see it your way. There are also issues that are based on public opinion at the time, I think they can be in march and one other month, but if you happen to have opinion in your favor if/when these come up, the issues can instantly change to be more liberal. Of course, if Congress and the Supreme Court are all conservative, these changes will likely be quickly reversed.
Keep in mind that as gun laws get more liberal, it becomes harder to acquire the better weapons. At arch-conservative gun control laws, you can buy AK47s no problem. You might want to stock up on them, too, because as soon as gun control moves 1 notch up, you lose access to the AKs. The more liberal it gets, the worse the guns are in the gun shop, you lose the good/expensive options. Of course, there are plenty of ways to get your hands on better guns, and I'll let you explore for that.
tl;dr - Don't keep all of your non-expendable people in one place, do tons of raids before even-year novembers, cheese the legal system, shoot up lots of cops to stop raids, sleepers are your friends