Taking into mind that you're a bronzie, reconsider this: who do you rely on to carry the game? Other bronzies, who you cannot possibly trust to be skilled at the game? You play with 4 other players who are probably bad at most aspects of the game. I think your starting point is already mistaken and I would rather have you play a carry champion who is mechanically simple - Kha'Zix is a great starter, IMO, because he has room for error, unlike many others.
If you still want to play Rek'Sai however... you need to decide how you want to play her. If you want to play the usual tank build:
Runes: Reds+Quints should be some combination of AD and attack speed. Quints should always be attack speed, it's just the most efficient. Yellows armor, blues should mostly be flat MR. If you have enough rune pages and feel confident against magic damage in the early game, consider MR/LVL.
Masteries: 12-0-18, Keystone: Strength of the Ages. Possibly 6-6-12, or 0-12-18, in which case take Runic Affinity. For 0-12-18, Merciless+Dangerous Game are the only real options.
Item builds: If you want to play to the control style then you need Tracker's knife into Cinderhulk. Sightstone is excessive. Buy the refillable potion, Rek'Sai's innate sustain + it is enough for the entirety of the game. At lvl 9, you must have the upgraded red trinket. Preferably you buy red trinket immediately after getting Tracker's Knife. Tank items you should buy in most games: Randuin's, Cinderhulk, Locket. Ignore Locket if the enemy team only has 1 magic damage champ who doesn't build a whole lot of it. Buy Merc's against CC heavy teams regardless of damage type. Against teams with only 1 AD, you only need 1 other armor item. GA should be built very late into the game.
Playstyle: Rek'Sai in particular has nasty gank routes, because her tunnels allow her to reach very far. You know the big wall in top lane on the left side of the river? From the river brush, you can easily get behind the enemy laner. Practice this a few times in custom games if you will.
When clearing jungle camps, you should not have any issue with HP.
In teamfights, you're the tank. Your job is to waste the enemy team's time, not solo kill anybody. The only times you look to kill enemy carries, is if you have somebody who can help you do that. Examples include: Fizz, Zed, LeBlanc. If someone is really fed, you help that guy more than anyone else. Remember, you are not the carry. You help others carry you.