I use a drain snake thing, as above on the same principle you need sturdy wire (eg wire hanger) and something 'catchy' on the end, like velcro, helps. This attaches to hairs and whatnot down the plughole.
Now, what kind of plug is it? Having put cleaner down it already I'm assuming it's not one which has an attached cover or you already detached it - but if it does and you haven't then that unscrews really easily by turning it. That then exposes the plughole completely. Otherwise in standard plugholes you can often pull them up to full expose.
If not, the wire stuff should still work. This is the main issue really, as bath plugholes tend to get blocked with hairs mostly.
What type of cleaner did you use? Mr Muscle has worked wonders on all manner of issues we've had over decades, hairs/fats etc (ick). It's more costly as the whole bottle goes in one use but it's great.
Also don't just tackle the bath - also go for the sink in the bathroom (assuming there is one). We once had a drainage issue and it turned out that both the sink plughole and bath plughole had to be cleared in order for the bath to drain properly. There were connected somehow.
Finally, get loads of cheap bicarbonate of soda and vinegar. Chuck down some soda, then some vinegar, it will all bubble up then go down again. Leave for 5-10mins, pour a kettle of boiled water down the plughole. Repeat. And repeat. And repeat. And... etc. I had issues in one place I lived (previous tenants 
) and used a pot of bicarb and a bottle of vinegar on the kitchen sink. As in all of it, and it wasn't as bad as your bath sounds.
However I'm not expert so I'm assuming industrial amounts of bicarb and vinegar won't massively ruin anything but I can't make any guarantees - Google! It's fine for regular cleaning.