So... how do you earn more points in arena?

I got to tier 20 as a ftp with no problem. Just use a team of 5 stars with full builds (just not low bst units), make sure you have a bonus unit, and win 7 times without deaths.

5 star your bonus unit.

Alfonse might suck, but if he's not at 5 stars and holding a full set of skills, you're losing potential points.

Also, the most important part. You need to do more than win. You need to make absolutely certain that you don't lose a single unit.

Eventually you're going to need heavily merged units, but getting to rank 20 shouldn't require that, only staying there. Flitting back and forth beween 19 and 20 is quite achievable quite quickly.

Take into consideration of what everyone is telling you, BST is also important as well as expensive skills that normally cost a bit more, such as Galeforce and Aether for specials, dual rally skills, passives that cost a total of 420 SP (630 if inherited from another unit).

Merges and rarity play a big part in score(always try to have 5* units, 4*+10 can work if you need to go that route, but you should always work toward 5*)

Sp cost(including seals)(It does go by original cost, not inherited cost). You always want to have the highest skill available(usually 3 like hone spd 3 for example). Always try to have the follwing

Weapon: Either a prf or a refined if available
Support: Dual Rally is worth the most(movement skills help with positioning but aren't worth as much)
Special: Galeforce or Aether(they are worth the most, otherwise any 300 point is fine)
A: Close Counter and Distant counter are worth 300 sp, otherwise 240 sp skills
B: 240 sp skill(some heroes have a 300 sp skill here Gunthraa for example)
C: 240 sp skill is the max
SS: Some seals are worth more than others at max(Atk ploy, Atk Smoke, Close Defense, Distant Defense, Deflect Skills, Guidance, Heavy Blade, and Panic Ploy are all worth 240 sp)
That is not to say that 200 sp skills don't work, you just lost some points doing it. You can look at the passive skills list for sp cost on skills.

Always make sure you are at lvl 40 in arena

BST does play a part, but it is only an extra 1 point for every 5 points in BST.

If you really want to get higher in Arena I would focus on heroes that you can get merges for easier and cover each others weaknesses well. Usually a Red/Green/Blue core leaving a spot open for a bonus hero that comes up. For Bonus heroes there is a rotating list that involves the askr trio and the free rotation map heroes.

Alf/Subaki/Wrys
Sharena/Olivia/Donnel
Anna/Lissa/Stahl
Alf/Gunter/Sophia
Sharena/Cecilia/Virion
Anna/Felicia/Hana

I would pick one unit from each set of 3 to work on for your bonus unit as they will always be in the rotation for now.

Use mainly melee units, or the new ranged with powercrept BST. Try to use legendary weapons if possible, and more expensive skills are better. Merged units also help, even if they are super-merged 4*.
Also try to keep your units around the same individual score, as it will decrease the variation in your battles.

This kind person made a spreadsheet that can calculate your arena score (Roughly). It also has a list of what actually affects the score.
https://www.reddit.com/r/FireEmblemHeroes/comments/75pf47/arena_score_estimator_a_spreadsheet/

I'm f2p, and I have 8 Crowns (the max now is 10 for players 100% t21), with six weeks in a row in T21 now.
What you need to make more points:
- High BST units: my core team is 3 armor, 2 with 170 BST;
- High SP cost for all units: Inherit don`t count more points, so, you need a special 500 sp for all the 4 units (as bonus unit, you can use Aether, Galeforce on Askr Trio), duo rally (like def/res), Distant Counter, and others - always at least with 240 sp. This is very hard, and fight without Reposition is a pain xD
- Merges: the core team, must be 5* +10. Choose your team, and merge them! I have Gwendy, Sheena, Draug (core arena), and Olivia, Effie, Cecilia and Rein, all +10!
-Buff/ Debuff and bonus: you need to kill using bonus unit, so, is a good idea to run all possible ploys in the core members, with buffs to!
- Bonus: use at least, 5* bonus unit!