Which gods are suitable for the Hard Carry achievement?

Every god is eligible to get that achievement. As long as you get 20+ kills in conquest.

Which gods and general strategy should I use to maximise my chances of obtaining this achievement?

In my eyes every god in every role except support role is suitable to get this achievement. In my eyes the easiest gods are the hyper carries e.g. hunters, assassins, certain mages (such as He Bo, Chronos, Janus, Sol etc.), But also warriors can get it for example Hercules, Guan Yu etc.

And for the best strategy that really depends on a number of factors mainly the god you are playing, also the match (for example if you are ahead you can play a bit more aggressive but if you and/or your team is behind you need to play a bit more defensive not to give the enemy even more of a lead) but also your playstyle for example if you tend to be more aggressive in the laning phase and forcing the enemy to make mistakes because they are under constant pressure or if you are a bit more defensive and letting the enemy near your tower and waiting for them to make a mistake by overextending and getting your teammate gank mainly just focussing on farming and getting yourself ahead that way. Those are both viable options. But the main strategy is (and maybe this is to plain and obvious) get fed.

As for me my strategy as mainly a solo laner is I mainly send them back to base or kill them when I am able to. After that I proxy one wave so they lose even more Gold and XP. If I am not able to send them back to base because they have a self heal I try to push my enemy under their tower and freeze the wave or push the wave under tower. That way they lose Gold and XP.

If I missed something feel free to post it in the comments and I'll add it.