Can you increase amps in your house with an external breaker?
There are 3 options I can think of, and I don't think any of them is a particularly viable idea for your situation:
Increase Utility Power
This is the normal way to do things. If you have, for example, 100 Amp service then you request an upgrade to 150 Amp or 200 Amp service. The utility will determine what is available and under what conditions. Along with this, you will need at a minimum a main breaker upgrade and very possibly a full panel upgrade. Which in turn may trigger updating other wiring due to code requirements. This is something that can only be done professionally and only with consent of the owner. The person listed on the electric bill, presumably the owner, is the one who would have to work out the details with the utility. This type of upgrade is beyond &quot;casual homeowner maybe without a permit&quot; land in most jurisdictions, plus since you are not the owner you couldn't do the work anyway, and typically a landlord can't do electrical work on his rental properties. So this equals $$$$.
Add another service drop
This is only slightly different from the above. The difference is that it would be a NEW connection, independent of the existing service. This is actually not often an option - more typically this would be for a second, separate, building that happens to be on the same property. This would require a new main panel (separate from the existing one) and utility agreement, and landlord permission, and a lot of work (cabling of new independent wiring for the new circuits). So this equals $$$$.
Separate power source
This would typically be a gas generator, but could also be solar or other sources. The problem is that these usually have either a high up-front cost (solar + batteries + inverter) or a moderate up-front cost plus fuel (gas generator). So this also equals $$$$.
All of these cost are likely to cost quite a bit. In the first two cases, this is really an investment in the property - if the landlord asks you to pay for the upgrades, you have consider how that will work for you given that your lease is not forever. The last option might allow you to take some stuff with you, but has its own problems (gas and maintenance for generator, high installation costs for solar).
If I were you, I'd start looking for a rental where you can get more power.