The slug cruiser is actually a fairly easy ship to win with, if you get over the initial hump. Use your dual lasers and, if pressed, the breach bomb to disable the enemy's weapons (a one-two shot with dual lasers will knock the shields down on early enemies and do damage to their weapons system), then immediately fire the anti-bio beam at the enemy crew (red dots) revealed by your Slug crewmembers' telepathy.
The goal is to kill the enemy crew before their ship explodes due to damage for maximum scrap. Obviously you must never use auto fire; timing is everything - the anti-bio beam will do nothing if the enemy's shields are up. Disabling the medbay on the enemy ship may be necessary, in which case a quick breach bomb should do the trick. If faced with an AI ship, destroy it if you have enough damage output to kill it without taking significant damage or get the hell out if you don't. You can also use the breach bomb to damage enemy crew, using the beam and the bomb at the same time should 1-shot most enemies.
Slugs' telepathy will provide you with a fairly good view of the rooms on your ship despite the lack of sensors (they see rooms adjacent to the one they currently occupy). If you believe there's a fire, but don't see it in the visible rooms, you can either send them to recon the ship or just vent the rooms you don't have sight of - slug repair gel will heal any hull breaches, allowing you to quickly regenerate lost oxygen after the fires are out.
Build-wise, you will want to find some extra crew as soon as possible, as well as upgrade your weapons and shield (standard procedure, really). Getting level 2 sensors will obviously help. Weapon pre-igniter is a good augmentation, because it allows you to disable enemy weapons before they have a chance to attack, and the scrap arm will make you even more obscenely rich.
For tips on how to deal with the boss specifically, see this question, but the most important thing to remember is that wiping out the enemy crew is the worst idea when fighing the boss, you should leave at least 1 of them alive.