Guild Wars 2

Can you only get achievements from this tab when you purchased Guild Wars 1?

Adine.2184:

Yes . HoM achievements require you to have gotten achievements in GW1 and put them in your HoM there to unlock items.

Almost this.
These rewards are a reward for showing dedication in the previous installment off the game. So you need to own atleast one campaign to have the first 3 hom- points (this will not lead to an achievement in gw2, but will unlock some armor skins).
You also need to link your gw1 account with your gw2 account.

To obtain more points you also need to have the expansion Eye of the north (see note) and actually play the game (though the first few points are easy to obtain and require little effort).

To get all points you need to own all three campaigns and Eye of the North expansion and play an extend amount for that game. The difficulty of reward is increasing. So the first points are easy, but the last ones take time and effort.

(note: it is possible to get up to 8 HoM points without owning the hall of monuments if you have the campaign called prophecy’s, this is however extremely difficult).

Skydown.6139:

Can you only get achievements from this tab when you purchased Guild Wars 1?

What tab?

Yes . HoM achievements require you to have gotten achievements in GW1 and put them in your HoM there to unlock items.