How to you access Undertale Game Files on a PS4

The Ruins save points don't give a kill counter, only later ones do.

You need to keep killing more monsters until battles give the message "But nobody came..." This must be done before approaching the boss monster for the area (at the point where you would normally fight them), even if you could normally turn back without fighting them. As long as you're on the Genocide route, save points checked outside the Ruins will give you a count of the number of monsters you have left to kill in the area. The count of monster kills for each section, followed by the name of the area boss, is as follows.

• Ruins: 20 kills (Toriel)
• Snowdin: 16 kills (Papyrus)
• Waterfall: 18 kills (Undyne the Undying)
• Hotland/Core: 40 kills (Mettaton NEO)

You don't need to kill every enemy you encounter, you're allowed to flee from or spare enemies, except for a select few), but you must also kill every unique enemy you encounter. This includes Shyren and Snowdrake. Although the Shyren encounter will happen even if the kill counter is depleted, the Snowdrake encounter will not and can ruin a genocide run.

Note that the requirements is in the number of kills, not battles, so battles with more enemies are more efficient for quickly reaching your goal. You will know that you have enough kills when interacting with a save point simply produces the message "Determination". If you ever leave the Genocide route, there's no way to get back on it without reloading a save from before you left the route.

Source: http://undertale.wikia.com/wiki/Genocide_Route#Requirements

There are basically 3 way that games on Steam can track your progress between PCs.

Steam Cloud

If a game has been made to use this feature, Steam games can store their saves to Steam's save cloud. Steam will then make sure that the use will have their save data remain in sync between different PCs. You can see if a game uses Steam Cloud by checking its store page, or your Library in List View mode if you already own the game.
Undertale does use Steam Cloud, despite the feature not appearing on the store page.

Steam Achievements/Leaderboards

This is quite rare, but some games use Steam Achievements or Leaderboards to track how far you've made into the game. There are only a few cases where this is even viable, but I know at least one example that opts to use this in stead of other syncing methods.

Third Party Storage

Most often seen on online-only games, like MMOs, some games track your progress by tying it to an external account. You'll have to log into this account separately, because this is the only way for the third party server to identify you. Although it may be possible that they could identify you with your Steam account, using your Steam account as a log-in substitute.