When you first recruit them, let them sit in a city with an Academy or better. They'll get the "Trained" attribute in a year or two, and stand a good chance of acquiring retinue such as translators. You can also send them off to a battle zone, where they can pick up the Foreign Hostage retinue.

Then negotiate lots of ceasefires, trade rights, and/or alliances (giving gifts doesn't seem to help, sadly). Presumably you could even provoke wars with a minor foreign power (blockading a far-off port, maybe?), just to then negotiate ceasefires and boost your diplomat, but this might bring down your global "trustworthiness" level.

Finally, just let them get old and learned: they'll gain even more influence as they age, although they'll also get slower and more prone to assassination.

Note that your initial diplomats are often hard to match in the later game, precisely because they're the ones negotiating a lot of ceasefires, trade rights, and alliances. You'll also see this with AI diplomats, where their initial diplomats become almost impossible assassination targets, just because they've negotiated so much and have such high influence. Later on there's just less opportunity for negotiation because all the deals have already been done, so it's harder to build up great diplomats. Still, if you start out lucky (with e.g. a diplomatic genius), get trained, and throw in a couple of retinue, you can be at +7 or +8 influence before he's even gotten old or negotiated his first trade deal.