I thought this game was fun enough to go for most of the achievements, but the bulk death achievements made me groan. The reminder that I'd died so many times was simply disappointing. So, after beating the game with around 250 deaths I made it my goal to finish all the other achievements except the 1000 death one.

First of all, you'll want to make your way through the campaign. You'll get most of the achievements that way, as well as set a reasonable goal for which death achievements to avoid. If you get the 1000 death achievement in campaign, this is simply a guide for completing the rest of the achievements most efficiently.

Each of the Arenas you make your way through in campaign has an arcade mode. I would recommend completing the entire campaign before grinding out the rest of the achievements in arcade. There are three different sets of achievements to get in campaign, for breaking a number of barrels in a single run, saving an amount of money, and purchasing some number of cards. You'll have to break 150 barrels in a run, save 5000 coins, and purchase all the cards. You earn coins during your run, which you can use to purchase cards from the shop.

But wait! What the game doesn't tell you (until it's too late) is that once you purchase more than four cards, you'll be locked into taking only four with you on a run, being able to spin for them with "Madame Zosma", with each spin costing an increasing amount of gold, or break the barrel to fill all remaining slots randomly.
In a context where efficiency matters, this is very antithetic to your goals. Selecting your best cards will cost you 275 coins which you'll then have to make up in your run, and leaving it to chance will run the risk of choosing worse cards. So, you basically have just four slots to purchase cards for. Stop at four, and you'll never encounter Madame Zosma and losing 35-200 coins a run. So take a deep breath and don't start saving just yet, you'll want to purchase the following four cards, in order, as soon as you can save up for them.

Your very first purchase, unsurprisingly, should be Greed. Unfortunately, you have to purchase two tier 1 cards before you purchase a tier 2 card, and tier 1 cards just aren't very good. In this context, it's worth purchasing the one that will amp up your coin production, even though it won't help you get to 150 barrels.

This is probably the most controversial recommendation I'll make. You've already purchased Greed so that you can get those coins fast, but what now? It still has to be a tier 1 card and you don't have many options. If something is calling to you, it's probably fine. Otherwise, I recommend Athlete.

Speedy movement can be a little distracting when you move in and out of it, but I think speed is essential in Penarium. Assuming some level of competence, spending less time to break barrels means less chance of death per barrel.
On the other hand, maybe you're very skilled, and have managed to make it through the campaign without triggering the "This is getting awkward..." achievement at 50 deaths. Then, I might recommend the Magnet.

You clearly don't need any help platforming, and the Magnet will improve your efficiency that little bit which could make the difference for you. Unfortunately, each card is only active for a little bit and takes time to build, so there's no chance for Magnet and Greed to combo, as I'd hoped when they were my first two purchases.

Barrels is massively undersold in both the text and card cost. This tier 2 card will spawn all the barrels at once, allowing you to blitz three or so levels on a single card. No other tier 2 power comes close. The next is probably portal, but since the cards are only active for a limited period of time, barrels just has more value as it can carry you through more stages.

The final card you'll want to purchase, and the best powerup in the game, is, of course, the Shield.

Unlike other cards, the Shield sticks around until you are hit, and then prevents you from dying. It essentially gives you another life, though you can't fall off the screen (or drown?). As such, it's extremely valuable, both from an efficiency perspective and for getting to 150 barrels.

Now you just need to collect coins and break 150 barrels in a single run. I found the second more difficult, but if you're finding the first more of a slog, you'll want to make sure you save enough for all the cards before purchasing any more. Personally, I got the 250 deaths achievement right towards the end of the campaign. I made this guide so that someone better than I might avoid getting it at all. Hopefully you find this guide useful, or at least amusing.
If you've gotten this far, there may be one last death-related achievement that's not a bulk death achievement that you're missing. 'Brainfreeze!' requires that you get hit with an icicle spike in midair. Hope you have an extra death left!